Population affects production which pretty much affects everything. 

Production = 2 * pop ^ 0.7

Your production then is allocated to wealth, manufacturing, and research to set your raw research and manufacturing. 

Your raw manufacturing is multiplied by your bonuses.

they say there are no stupid questions so here goes.

What is ^ ? I assume it is a symbol used in place of something not on a standard kb

^ means to the power of doesn't it so 2*pop*(0.7*pop) essentially.

Also pop affects morale. The more you go over the limit of your morale the lower your morale goes.

Think of your calculated morale as 100%. Each .1 of population over that morale is 1% unhappiness.

That's actually an interesting point. There must be a value at which an extra  farm diminishes below the value of an extra factory when calculating the overall manu of a planet. same deal with wealth and research.

anyone worked out roughly what that point is yet?

As Azunai_ says, the math isn't exactly simple for this. You generally have to sacrifice two production buildings per farm, and moreover since population affects production and factories multiply production, you can have multiple points at which it's better to build a (farm + approval structure) than two factories/labs/markets. Additionally, there's a question of time. Farms don't instantly increase your production; you have to build up to it. Thus, in the short term, it will most likely be better to build an output structure than a farm; in the long term, the picture is a little more messy.

At any rate, you can check whether a farm or a factory is better in the long run testing if

(B + 2 * (1 + F + f)^0.7 *  (P + p)^0.7) * (1 + M) > (B + 2 * (1 + F)^0.7 * P^0.7) * (1 + M + A)

where B is the flat production of the planet, F is any percentage bonus to food production that the planet has, f is any percentage bonus to food production that adding your farm will add, P is the planet's maximum population without the farm you're considering adding, p is the population you'll get by adding the farm, M is the planet's overall output bonus, and A is the increase in the output bonus you'll get if you build output boosting structures instead of farms and approval structures. Express all percentages as decimals, and note that I'm assuming in the above formula that you add enough morale structures to keep the approval rate constant; if you don't, this gets messier because the approval rate has a modifier to output that stacks multiplicatively with the output bonus from structures/planet abilities/etc, and I believe that the approval modifier is a piecewise linear function defined by a set of points in the XML rather than something easier to work with in the formulas. If the LHS is greater than the RHS, then you're better off with the population in the long run, while if the RHS is greater than the LHS, you're better off with the factories/labs/markets. M can be calculated as

M = [manufacturing fraction] * [manufacturing bonus] + [research fraction] * [research bonus] + [money fraction] * [money bonus]

A further complication is that these points do not remain constant for all farm types, factory types, and total levels to each type of structure.

I personally would suggest not bothering with farms on most planets; at +25% to the desired output type per factory/lab/market and +2 maximum population per farm, you're not going to see any long-term benefit to adding a farm unless you have more than 10 tiles to play with, assuming you need to build 1 approval structure per farm, and you're going to fall behind in the short term by building farms instead of factories/labs/markets (though there could be a condition, say if you're building a purse world with its base production, where you'd have a short-term benefit because the population reaches the base maximum population and then the maximum population with the farm included before you complete all the markets, which means you'd finish the build up period somewhat faster, letting you switch to the chosen output type sooner; I wouldn't concern myself terribly much over this, however).

Using solver in excel ignoring dimension ignoring adjacency. You basically need 1 farm per factory. You do have to account for approval and other buildings. For the most part it gets the job done if you don't want to pull a calculator out. Approval buildings get a flat 1 increase per adjacency bonus and farms get .05. So it's actually not 1:1 farm to approval building. 

So what does it mean when a bonus tile gives, say, +2 pop for structures built on top of it?

Which is it? Don't build farms on <10 colonies and only in late game, or one per factory, which seems kind of extreme?

cool that's about what my intuition thought. I was working off 25/75 roughly. I think by the time you factor in production bonuses etc this will be true.

generally I use the main landmass for factories, stacking as much adjacency as possible, and remaining landmass for farms, trying to achieve about 25/75. As a very rough approximate. 

[Assuming I am not running any projects].

If one checks the output boxes on the bottom left of the colony screen, you will see the bonuses listed for manufacturing, research, and wealth.  Also you will see Raw Manufacturing, Raw Research, and Income in each respective box.  (Why the last isn't called Raw Income I don't know).   If you take the raw numbers and multiply by 1 plus the bonuses for each output you will get your actual production.   Furthermore, wherever you put your cursor on the production wheel, the TOTAL of the three raw productions will be the same, and it will be split according to how you allocate to the three types of output.   Easy enough, every time I have checked it is spot on.

So the only real trick is to figure out what the actual "raw production" is.   There is a big problem with distinguishing between manufacturing and production, which are quite different.   Production is the raw output, manufacturing is how much manufacturing you get.   Confusing at best.  But not only that, there is such a thing as "Construction".  Which is sometimes Production.    

Illuana has given a formula based on the entries in the file GalCiv3GlobalDefs, which gives the exponent as 0.7 and the multiplier as 2.  

First, the raw production is affected by a multiplier that depends on the approval (the approval to production curve).   At 100%, it is +25%, which I mention because that is the example I will show.   Other values are listed in the GlobalDefs file.   So this multiplier needs to be applied after the above calculation.  So at 100% approval I guess that

Production = 1.25 * (2 * pop ^ 0.7  + bonuses to raw production)    (FYI,  at 50% approval there is no bonus, and below that there is a penalty or negative bonus).

Second, there are various raw production modifiers.  For example, a colony capital gives a flat +5 to total production.   A Durantium refinery gives a flat +1 to total production.   A racial trait, productive,  gives a modifier to the raw production, depending on whether you take the first or second choice.  A starbase factory at level one gives +1 Construction.  And this latter does factor in because if you look at the raw output of a colony and destroy a nearby starbase factory, the raw production will decrease.  In my observation it decreased by 1.25 per factory, suggesting it was being multiplied by the approval factor.

For example, from a save game I check a colony that had population of 4.0.   It has an approval bonus of 25%, a colony capital bonus of +5, and a  Durantium Refinery bonus of +1.  (Curious that if I destroy the refinery I lose 1.5 to raw manufacturing instead of 1.25 as expected. Oh well).    There are three lvl 1 factory starbases  nearby.   A racial trait of +1 to production (good question as to whether this gives a +1 to Raw Production as well as +15% to manufacturing...I am thinking it does!)  gives a +15% bonus to manufacturing, and it has a reported Raw Manufacturing at 100% manufacturing of 19.1.  Question is, how to get from this data to 19.1.   

My guess would be 1.25 * (2 * 4^0.7  + 3 + 5 + 1 + 1) = 19.09  That's pretty close.  (3 SB factories, a colony capital, a +1 production trait, and a Durantium Refinery).   But problem is if I destroy the starbases and the refinery the calculation is now wrong:  I get 14.1 instead of 15.3 as shown in game.

My question is whether any of  this is correct and my suggestion is that things ought to be labelled much more clearly.  

I think it is about time for someone to give the accurate formulas that calculate the Raw Production.  Never mind the effect of projects, which is a different can of worms.

FYI the system example:  

But that is only in the short term... I agree that in the short term you are mostly better of with pure enhancement improvement. But games can last many hundreds of turns, even well more than a thousand turns if you don't use that victory condition. After a certain point you will gain allot more from population improvements rather than pure bonus enhancement buildings.

Looks like you are right...the econ ring does give 10% production bonus (not manufacturing).   And I also miscounted the starbases, as one of them is a military base!  Ouch.

That would make my calculation

My guess would be 1.25 * (2 * 4^0.7 + 5 + 1 + 1) * 1.2   where I have dropped the +3 for starbases but added two 10% multiplicative additions, which is a factor of 1.2.

This gives 18.4 which off a bit (19.1 being the raw production stated in game).  I get 18.6 if I do the 10%s sequentially (1.1 x 1.1) but that doesn't help much.

Thanks for the catch at least I am a little closer now!