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:

Research Rate Change

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To derive the equality condition, take "Change in Research Rate" = 0, and add 1 to both sides:

Continue manipulating:

Now, note that

Substitute these expressions to get:

And, finally:

Equality Condition

This simplifies for large (CountrySize ≥ 8) countries to:

Therefore:

Equality Condition for Large Countries

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.

Research

Research:

Research (Advanced) • Research Math Proofs • Research strategy • Research Cost - Supporting Data ( and )

Technology:

Technology tree • Technology groups • Westernisation • Country size modifiers