Kolpo 's right. The marginal cost analysis that I did above misses two things. First, each stock market you add makes all your existing population worth more - so if you fill two squares with stock markets then you face a different formula when figuring out what to do with the next two squares. Second, the total number of squares is limited - you can't keep adding stock markets forever, you have to maximize the revenue from the available tiles.
So let's break out the differential calculus and actually figure this thing out. Let's make the following definitions:
PQ = Number of tiles devoted to making money (not research or production) on a given planet. Usually less than planet quality, because you'll build a Starport and some factories even on your tax world.
f = Number of farms on the world. We assume we need 1 morale building per farm, so the number of tiles consumed is 2 * f.
m = Number of stock markets on the world. Each consumes 1 tile.
T = Total tax revenue on the planet.
To simplify the formulas, we'll assume that base food is 5 (i.e. this is not our homeworld) and that the empire-wide tax rate is 50%.
The tax generated on the planet is calculated as
T = (5 + f * 8) Billions * 10BC * 50% * (100 + m * 30%)
Our goal is to maximize T.
We also know that the total number of farms, morale buildings and stock markets must equal the available tiles, or
PQ = 2 * f + m, or m = PQ - 2 * f
Substituting this into the main formula, we can express the tax revenue in terms of planet quality and number of farms
T = (5 + f * 8) Billions * 10BC * 50% * (100 + (PQ - 2 * f) * 30%)
Multiply everything out and simplify, leaves us with
T = 25 + (7.5 * PQ) + (25 * f) + (12 * f * PQ) - (24 * f ^ 2)
Take the first derivative, to find the rate of change in taxes with respect to number of farms
dT /df = 25 + (12 * PQ) - (48 * f)
Taxes will be at a maximum when the first derivative is 0 (when a line drawn tangent to the curve is flat). So
0 = 25 + (12 * PQ) - (48 * f)
Finally, this tells us the optimal number of farms to build on a given planet
Fmax = (25 + 12 * PQ) / 48
For each farm, build 1 morale building and fill the rest of the squares with stock markets.
If we apply this formula to some examples, we'll see that we reproduce the values that
kolpo got in his table. Say that we have a Planet Quality 21 world that we wish to be a money planet. We'll build a Starport and two factories, leaving 18 squares to make us some money. That means PQ = 18 in our formula.
We should build (25 * 12*18)/48 = 5.021 = 5 farms
We'll also build 5 morale buildings
That leaves us 18 - 2 * 5 = 8 stock markets
We can expect this world to generate T = (5 + 5 * 8) Billions * 10BC * 50% * (100 + 8 * 30%) = 765BC when it reaches it's maximum population.
Here's a table of the breakpoints in the optimal number of farms to maximize revenue:
| PQ (Free Tiles) | Farms |
|---|
| 10-13 | 3 farms |
| 14-17 | 4 farms |
| 18-21 | 5 farms |
| 22-25 | 6 farms |
| 26-29 | 7 farms |
You might want to note, though, that it's gonna take a long time to fill up a planet with 7 farms. You might not have that many turns left in your game!
