# Difference between revisions of "Gotikiller's Capitalist Theory"

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As someone who loves filling factories, I have always wondered if it was possible (or more accurately... ''feasible'') to concentrate enough pops in a state so that the rate of population increase is large enough for there to be 5 factory workers rearing to go every time a factory is completed. This theorem proves that the number is not completely unfeasible and provides an equation for finding the self sufficiency critical number of a province based off of it's growth rate. | As someone who loves filling factories, I have always wondered if it was possible (or more accurately... ''feasible'') to concentrate enough pops in a state so that the rate of population increase is large enough for there to be 5 factory workers rearing to go every time a factory is completed. This theorem proves that the number is not completely unfeasible and provides an equation for finding the self sufficiency critical number of a province based off of it's growth rate. | ||

− | ====Finding the | + | ==== Finding the needed population ==== |

Lets find what is the minimum amount of population at which sufficiency is reached. First here are our assumptions: | Lets find what is the minimum amount of population at which sufficiency is reached. First here are our assumptions: | ||

Line 15: | Line 15: | ||

*Capitalists build at every opportunity they can | *Capitalists build at every opportunity they can | ||

− | First since this is a geometric sequence, lets take a general formula for a geometric sequence and apply it to this scenario:< | + | First since this is a geometric sequence, lets take a general formula for a geometric sequence and apply it to this scenario: |

− | Since we need to fill 38 levels of factories over two years (hence t = 24 for 24 months), we need to increase the size of our craftsmen by 38*40 | + | :<math>g_{t} = g_{0} \cdot r^{t}</math> |

+ | Since we need to fill 38 levels of factories over two years (hence t = 24 for 24 months), we need to increase the size of our craftsmen by 38 * 40 000 or 1 520 000. So lets solve this equation for the original population and then plug in our variables to solve specifically our solved general equation. | ||

First lets start with our original equation plugging in 24 for t | First lets start with our original equation plugging in 24 for t | ||

− | < | + | :<math>g_{24} = g_{0} \cdot (1+ r)^{24}</math> |

− | Then divide both sides by (1+r) | + | Then divide both sides by (1+r)<sup>24</sup> |

− | < | + | :<math>g_{0} = \frac{g_{24}}{(1+ r)^{24}}</math> |

− | Then since g24 is defined as g0 + 1 | + | Then since g24 is defined as g0 + 1 520 000 or in math parlance: |

− | < | + | :<math>g_{24} := g_{0} + 1520000</math> |

We can plug in g0 + 1,520,000 for g24 | We can plug in g0 + 1,520,000 for g24 | ||

− | < | + | :<math>g_{0} = \frac{g_{0}+1520000}{(1+ r)^{24}}</math> |

Then split the numerator of the fraction on the right side into two parts | Then split the numerator of the fraction on the right side into two parts | ||

− | < | + | :<math>g_{0} = \frac{g_{0}}{(1+ r)^{24}} + \frac{1520000}{(1+ r)^{24}}</math> |

Then subtract the fraction containing g0 from both sides | Then subtract the fraction containing g0 from both sides | ||

− | < | + | :<math>g_{0} - \frac{g_{0}}{(1+ r)^{24}} = \frac{1520000}{(1+ r)^{24}}</math> |

Then on the left side anti-distribute g0 from both terms | Then on the left side anti-distribute g0 from both terms | ||

− | < | + | :<math>g_{0} \cdot \left(1 - \frac{1}{(1+ r)^{24}}\right) = \frac{1520000}{(1+ r)^{24}}</math> |

− | Then divide both sides by | + | Then divide both sides by 1-(1/(1+r)<sup>24</sup>) |

− | < | + | :<math>g_{0} = \frac{1520000}{\left(1 - \frac{1}{(1+ r)^{24}}\right) \cdot (1+ r)^{24}}</math> |

− | Which, after you distribute in the (1+r) | + | Which, after you distribute in the (1+r)<sup>24</sup> in the denominator of the right fraction, leaves you with: |

− | < | + | :<math>g_{0} = \frac{1520000}{(1+ r)^{24}-1}</math> |

− | Since plugging in .25% into this equation yields you 24 | + | Since plugging in .25% into this equation yields you 24 612 577. I am able to say with a very large amount of certainty that if you have a province with a growth rate of .25% and a population of 24 612 577 or 616 craftsmen pops, it has become fully industrialized and will afterwards be a net exporter of populace and no longer requires immigration and the such to support its economic expansion since its pop growth is now sufficient to cover its own labor needs. |

− | its own labor needs. | + | |

+ | Just remember this is in a perfect scenario and in reality due to capitalists NOT building every single second that they can, most likely your critical number will in actuality be significantly lower (i.e. 50+%). The point of this critical number is to give the player a number where he can say with 100% certainty that the province will be industrialized. | ||

====Conclusion==== | ====Conclusion==== |

## Latest revision as of 10:11, 23 April 2017

## Contents

## Capitalist Theory 2.00 by Gotikiller

### Theorem 1 - The Existance of a finite Maximum Growth Rate

As the number of workers a reaches infinity, Capitalists will start expanding factories at every point in which they can eventually resulting in a constant rate of growth. This is because at maximum, since factories can only be expanded one level at a time, only 38 levels of factories can be expanded every two years (i.e. there are 15 factories that can be expanded every two years and 8 that can be expanded every year). As this limit is reached, assuming that labor is not a factor, then a state has reached its maximum economic growth rate of 38 factories/2 years.

### Theorem 2 - Provincial Labor Self Sufficiency

#### Introduction

As someone who loves filling factories, I have always wondered if it was possible (or more accurately... *feasible*) to concentrate enough pops in a state so that the rate of population increase is large enough for there to be 5 factory workers rearing to go every time a factory is completed. This theorem proves that the number is not completely unfeasible and provides an equation for finding the self sufficiency critical number of a province based off of it's growth rate.

#### Finding the needed population

Lets find what is the minimum amount of population at which sufficiency is reached. First here are our assumptions:

- The state in question is a one province state, with a growth rate of .0025 (.25%)
- Capitalists build at every opportunity they can

First since this is a geometric sequence, lets take a general formula for a geometric sequence and apply it to this scenario:

Since we need to fill 38 levels of factories over two years (hence t = 24 for 24 months), we need to increase the size of our craftsmen by 38 * 40 000 or 1 520 000. So lets solve this equation for the original population and then plug in our variables to solve specifically our solved general equation.

First lets start with our original equation plugging in 24 for t

Then divide both sides by (1+r)^{24}

Then since g24 is defined as g0 + 1 520 000 or in math parlance:

We can plug in g0 + 1,520,000 for g24

Then split the numerator of the fraction on the right side into two parts

Then subtract the fraction containing g0 from both sides

Then on the left side anti-distribute g0 from both terms

Then divide both sides by 1-(1/(1+r)^{24})

Which, after you distribute in the (1+r)^{24} in the denominator of the right fraction, leaves you with:

Since plugging in .25% into this equation yields you 24 612 577. I am able to say with a very large amount of certainty that if you have a province with a growth rate of .25% and a population of 24 612 577 or 616 craftsmen pops, it has become fully industrialized and will afterwards be a net exporter of populace and no longer requires immigration and the such to support its economic expansion since its pop growth is now sufficient to cover its own labor needs.

Just remember this is in a perfect scenario and in reality due to capitalists NOT building every single second that they can, most likely your critical number will in actuality be significantly lower (i.e. 50+%). The point of this critical number is to give the player a number where he can say with 100% certainty that the province will be industrialized.

#### Conclusion

To achieve the fastest rate of industrialization of a state, one must try to reach the given critical number for your growth rate. This achievement, through pro-immigration policies and forced migration of farmers by the state (by converting them to craftsmen and taxing them until they can not afford life goods, causing them to migrate), should be a paramount objective of a player since economic power yields in the long term Military and Prestige which when combined, at least in Victoria, yields Victory.