To get a rough idea of what is needed to bootstrap a given colony, find the manufacturing percentage of that colony.
Multiply that by the pop growth percentage.
Then multiply by the population (in millions).
That’s the number of workers (in millions) per year that colony is creating.
(Sort of. It’s usually actually less than that, because the manufacturing percentage is decreasing as the pop grows. We’ll get into that later.)
Multiply that by 20.
That’s the number of (typical) production installations needed per year for the new workers.
Multiply that by 120. That’s the cost of building those (typical) production installations.
Using a bit of algebra, we can redefine things as follows:
mPop * manupct = mWrk (millions of workers)
mWrk * growthpct = new mWrk (millions of new workers)
new mWrk * 2400 = new confac cost
If we assume that your worker pool is currently working only in confacs (i.e. you have no other jobs at this colony):
BP per year = mWrk * const rate * 20 (const rate is the construction rate tech you have researched. starts at 10 with TN tech.)
…which is the same as:
mPop * manupct * const rate * 20
To make enough BP per year to build confacs fast enough, this must be true:
BP per year >= new confac cost
… mPop * manupct * const rate * 20 >= mPop * manupct * growthpct * 2400
… const rate * 20 >= growthpct * 2400
… const rate >= growthpct * 120
Now, growthpct depends on population size and governor/sector bonuses (and racial pop growth rate, and crowding penalties, but let’s ignore all of that for the moment).
The growthpct before applying bonuses and racial pop growth is the smaller of 10% or this:
20% / cube root of mPop
Simple math tells us that the growthpct is at the 10% cap until the population reaches 8m.
For capped populations (pop <= 8m), and before accounting for gov/sec bonuses, to provide new jobs fast enough to keep up with pop growth:
const rate >= growthpct * 120
… const rate >= 10% * 120
… const rate >= 12
For below-cap populations (8m+ pop), the const rate you need is
(24 / cube root of mPop).
In either case, if you have a gov/sec pop growth bonus, apply that to the number you get.
This tells you the actual const rate you need.
If you tend to focus on pop growth governors, you might need to be 3 or more tech levels up the const rate line before your small colonies can build confacs fast enough to keep up with pop growth.
At starting tech levels, colonies with a good governor can be hard to keep up with even well past the 8m pop threshold.
E.g., consider a 22m colony with 40% pop growth bonus
24 / 2.8 * 1.4 = 12
You’d need to be at const rate 12 just to keep up.
And keep in mind: all of the above is assuming all your workers are in confacs and you are devoting 100% of the queue to new confacs (or other typical production installations).
But unless the colony cost is zero, you also need infrastructure for your growing colony.
If you are building that infrastructure locally (and you should be; it is far better to build infra locally and ship in the confacs than vice versa), you’ll have less BP available for installation construction.
The exact amount depends on the colony cost. Details thereof are beyond the scope of this post. Suffice it to say:
As col cost increases from 0 to 5, the time needed to bootstrap increases, and the benefits decrease.
Above col cost 5, bootstrapping is a fool’s errand unless the colony is going to remain small.
The higher the col cost, the smaller the colony must remain in order to be a reasonable endeavor.
In practice, I usually build nothing but infra on a new colony until I have enough infra for 10m pop.
Then I switch to building just enough infra to keep up with pop growth, and building confacs with the remaining BP.
This won’t keep up with worker growth for quite a while, so I periodically ship in enough confacs for the available workers.
A final observation: bootstrapping is much, much, much easier if you supplement production with construction ground units.
But I’ve done enough math for one post.
Oh, and here’s the “we’ll get into that later” stuff about marginal growth rates:
The percentage of population allocated for manufacturing decreases as the pop increases, to a minimum value of 5 * (5-colcost) at 240.1m pop.
This means that multiplying your current percentage times your annual rate of pop growth will give a slightly higher number than you will actually gain over the next year.
In relative terms, the difference is larger for smaller colonies.
In absolute terms, I find that the difference does not make much of an impact.
As a new colony grows, I build infra until it reaches 10m pop, and then I build infra and confacs. I assign build queues for one year at a time.
When the colony is small, a large percentage of the queue is needed for infra. Little is left over for confacs.
The percentage needed for infra drops as the pop grows, because the pop growth rate drops in relative terms faster than the worker growth rate.
Each year, I have more and more BP percentage left over for building confacs, but it is still many more years before I can build enough to keep up with worker growth.
So, being a bit high with my estimate of the number of new jobs created in the next year just means that I’m doing slighly better than I think I am at keeping up each year.
It doesn’t change the plan, though. I keep bootstrapping as hard as I can until that boot is strapped.