Is there anyway to see how population is affecting manufacturing on the planet screen? I see a Raw Manufacturing value
The sum of your raw manufacturing, raw research, and raw wealth production is your raw production. Population adds a bonus to raw production which may be computed as
2 * (population)^0.7
This bonus adds to any flat production bonuses you have (there's a +5 from the colony capital, the homeworld gets another +5, there's a tech for +1, and I think there's a few other things, like an improvement or two, which add flat production to your planet). Thus, a world with a population of 10 will receive about 10 production from its population. There are also a few things which give percentage modifiers to production (approval and economic rings, mostly), which multiply the base production (i.e. the sum of the flat production bonuses and your population-based production).
Production is then distributed out over the three output types based upon your wheel settings to generate the raw (output type) numbers, which are then multiplied by the sum of the bonuses to that output type to produce your gross (output type) numbers.
So yes, if you want to see your production, you can set your sliders to 100% of some output type and then look at the raw (output type); that will be your world's production. There isn't an easier way to look at this at this point in time.
I'm assuming that under the right conditions, building a farm could result in more overall manufacturing than building another factory, although we'd have to know the exact equations in play to determine that break point. Is there any rule of thumb in this regard?
We do know the equations:
[Production] = (1 + [approval mod] + [production bonuses]) * ([flat production] + 2 * [population]^0.7)
[Manufacturing] = [manufacturing fraction] * [Production] * (1 + [manufacturing bonuses])
[Research] = [research fraction] * [Production] * (1 + [research bonuses])
[Net Wealth] = [wealth fraction] * [Production] * (1 + [wealth bonuses]) + [tourism] - [maintenance]
[tourism] and [maintenance] are not based on production. I'm not certain if a planet's population has an effect on tourism; the only two things I know about tourism are that there's a number in GalCiv3GlobalDefs called GalacticTourismSpendingMod which is set to 0.25 by default, and that there are buildings which modify it.
It will therefore be beneficial to build a farm if
[production with farm] * [old multiplier] > [production without farm] * [new multiplier]
or alternatively
[production with farm] / [production without farm] > [new multiplier] / [old multiplier]
where [production with farm] is the production you would have with the farm and [old multiplier] is the overall output multiplier you currently have, while [production without farm] is the production you currently have and [new multiplier] is the overall output multiplier you will have if you build (some kind of output multiplier building) in place of the farm(s) and approval building(s) you're considering adding. Note that you need to include the approval multiplier in the overall output multiplier or in the production numbers (pick one; don't include it in both), and that adding farms will generally (though not always) result in you wanting to add approval structures since the approval modifier stacks multiplicatively with the output multipliers, which can make it a fairly severe penalty (the approval modifier ranges from -25% at 0% approval to +25% at 100% approval and follows a curve defined by several points in GalCiv3GlobalDefs; open the file with a text editor or a web browser or some other program that can read the file and search for 'approvaltoproductioncurve' if you want to see what the points are, but it's approximately the cubic function f(x) = x^3 + 0.25*x + 1, where x = [approval] / 100% - 0.5, though I think the game actually uses a linear interpolation between the points listed in the XML file).
If you know the values, then you can use this formula to compute the value of [old multiplier] where it becomes better to add a farm than a new building (or buildings, if you include the approval structure you probably will need to add with the new farm):
b = ((B + 2 * P^0.7) * (1 + a) / (B + 2 * (P + f)^0.7) - 1) / (1 - (B + 2 * P^0.7) / (B + 2 * (P + f)^0.7))
where b is the bonus for the old multiplier ([old multiplier] = 1 + b; b*100% is the number reported in the output tool tip, though if you're using a world to obtain multiple production types you're going to have to compute the weighted average of the bonuses rather than just looking at the bonus to one output type), B is your flat production, P is your maximum population without the new farm, f is the bonus to maximum population you get by adding the farm, and a is the bonus you can add to your output multiplier by adding new output structures rather than the farm and any associated approval structures to maintain a constant approval rating.
For a planet with a base maximum population of 5 and a flat production bonus of 5, if a farm gives +2 maximum population and each output structure gives +0.25 to the output multiplier (+25%, e.g. a basic factory with no levels), then you will want to add farms when the output multiplier is +70%, and again when the output bonus is +113%, and again when the bonus is +155%. However, be aware that these were computed ignoring approval effects, which means you traded one farm for one factory (or other appropriate structure and vice versa), which is inadvisable.
