Research Math Proofs

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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:

GeneralProof1.png

We consider TechProgress here as a function of the size of a country (CountrySize). Thus:

GeneralProof2.png

We will proceed from this relation.

Now observe the impact of adding x provinces to CountrySize:

GeneralProof3.png

Expand the denominator as per the Technology Cost formula to make the relation to CountrySize explicit.

GeneralProof4.png

And expand the TotalResearchIncome[CountrySize + x] term:

GeneralProof5.png
GeneralProof6.png

So we have:

GeneralProof7.png

The rest is algebraic manipulation...

GeneralProof8.png
GeneralProof9.png
GeneralProof10.png
GeneralProof11.png
GeneralProof12.png
GeneralProof13.png

And, thus, we have:


GeneralProof14.png


Research Rate Change




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

GeneralProof15.png

Continue manipulating:

GeneralProof16.png
GeneralProof17.png
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Now, note that

GeneralProof20.png
GeneralProof21.png

Substitute these expressions to get:

GeneralProof22.png

And, finally:

GeneralProof23.png


Equality Condition

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

GeneralProof24.png
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GeneralProof26.png

Therefore:

GeneralProof27.png


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.