I'm curious what the exact formula that determines land cost is. I've gathered a bit of data so far to investigate it.
Linear/first order polynomial: This quickly gets ruled out as you include more data. (image 3 at bottom)
Exponential: Checking how much the cost increases per level, and then how much the increase in cost increases per level, and then how much the increase in the increase in the cost increases per level... suggests an exponential (or complicated power or high order polynomial) component. But, once enough data is included, the data appears to follow a curved path even with a logarithmic y-axis (image 1), so it must have a non-exponential component.
Second order polynomial: It appears to follow the trend line, but it's ruled out by the derivatives.
Power: This looks promising. With log x and y axes, it looks like a straight line, and the trend line follows the data very well.
Could the fine function of
give us the correct price of 1 additional square km. of land (p) when your land size has a given area (a)?
Predictions:
If you have 2,000 km
2 of land, the next km
2 would cost 11,903,273 currency.
The observed cost is 11,903,318 currency.
If you have 5,000 km
2 of land, the next km
2 would cost 117,511,038 currency.
The observed cost is 117,585,895 currency.
If you have 100,000 km
2 of land, the next km
2 would cost 2.0952 * 10
11 currency.
The observed cost is 2.1027 * 10
11 currency.
So it seems to be the correct formula, even though it must have more decimals.
By integrating from 100 (the land you start with) to your final land area, a
F, we can then find how much money was spent getting all the land:
So this says I've spent 788,000,000 currency on getting my total of 1,080 km
2 of land.
Aceziyora Kazerano has spent 385,000,000 currency on the 880 km
2 of land.
The Moltinian Empire has spent 34,700,000 currency on their 443 km
2 of land.
And the Ivory Coast with 2,000 km
2 of land has spent 6,800,000,000 currency on land.
And that the first square kilometer of land you buy should cost 6,760 currency. I can't remember the number anymore.
Anyway, here's a graph on how the total land cost goes as a function of your size.
The question is then, how does your income go as a function of your population, roughly? I have a bit of a feeling that the income from population isn't big enough to balance this land cost, which would mean bigger nations would tend to end up with higher and higher population densities.
