Other comments (and the answer) are correct, but you are dismissing them as not making sense. But I think you're missing part of the logic here.
At first sight, your observation is correct that production focus should focus more on production. However, you're forgetting about the impact that tiles with multiple resource types have.
For example, consider a city with 6 pops, and access to the following tiles:
- 2 Food 2 Gold (x6)
- 1 Food 2 Prod (x6)
- 3 Food (x6)
You need 12 food to sustain your population.
For example, if you were to put your city on a gold focus, it would assign all pops to the food/gold tiles. They make enough food (12) to survive and it also favors the gold. Total output: 12 food, 12 gold.
Now let's say you push your focus to production. You may initially think that the game would then assign all pops to the food/prod tiles, but that wouldn't work. You'd only be getting 6 food, not 12, because there is less food on the food/prod tiles than there is on the food/gold tiles.
To account for the loss in food, the logic will need to assign some pops to the food tiles, which has a dramatic impact: 3 pops on the food tiles, 3 pops on the food/prod tiles. Total output: 12 food, 6 production.
This isn't your exact scenario, but it's a simple example that showcases the point. By having the AI favor prod tiles over any other tiles, they are likely going to end up on tiles which generate less food in total, which means that some pops now need to be diverted to high-food yielding tiles to ensure the population doesn't starve.
When you get into the nitty gritty of pop assignment, you will see that the algorithm isn't perfect. This is why a lot of high level players tend to micromanage their cities, at least in a stage where city resources are scarce and a few points here and there can make a large percentage swing in city output.
The algorithm's imperfection can cause issues here. For example, consider that the city looks like this:
- 2 Food 2 Gold 1 Prod (x6)
- 1 Food 2 Prod (x6)
- 3 Food (x6)
Now, there is a better solution, the same as when you were on a gold focus. Putting all 6 pops on the food/gold/prod tiles nets you a total of 12 food, 12 gold and 6 production.
However, this might not be how the algorithm is built. For example, I can imagine a simple algorithm that does the following:
- Rank all tiles according to [focus] and assign all pops to the top tiles.
- While there is starvation, unassign 1 pop from the assigned tiles with the least food, and assign it to the tile with the highest food yield.
- Ignore all other unmentioned resources.
Of course, I have no idea what the internal Civ V logic is, this is just an example.
This would on average give you a fairly accurate assignment for maximizing your focus. However, it would lead to the initial solution of (3 pops on the food tiles, 3 pops on the food/prod tiles. Total output: 12 food, 6 production) and it will completely miss the solution of (6 pops on the food/gold/prod tiles. Total output 12 food, 12 gold and 6 production). This means you lose 12 gold without actually gaining anything in return!
Like I said, I don't know the exact algorithm used, and the example I used here has simple numbers for clarity's sake; but you can see how the assignment logic can miss things that seems obvious to humans, simply because the algorithm doesn't test for every possible combination. What I can conclusively say is this:
If tiles never had more than one resource, your observation would be mathematically and logically correct. However, when tiles have multiple output types, the additional resources can cause side effects which can have a dramatic impact on the outcome (the bigger the city, the more impact this can have on the final numbers).