Research Math Proofs
Back to Research Math.
We begin with a basic mathematical description of the rate of technology gain in a country, denoted here as the TechProgress term:
We consider TechProgress here as a function of the size of a country (CountrySize). Thus:
We will proceed from this relation.
Now observe the impact of adding x provinces to CountrySize:
Expand the denominator as per the Technology Cost formula to make the relation to CountrySize explicit.
And expand the TotalResearchIncome[CountrySize + x] term:
So we have:
The rest is algebraic manipulation...
And, thus, we have:
To derive the equality condition, take "Change in Research Rate" = 0, and add 1 to both sides:
Now, note that
Substitute these expressions to get:
This simplifies for large (CountrySize ≥ 8) countries to:
Note that in both the Equality Condition and Research Rate Change formulae, the marginal cases may be easily derived by taking x = 1, and making the substitution AverageProvinceResearchIncome = NewProvinceResearchIncome.