Economic Study: Tax Efficiency Tables, Corruption and Tax Rate

[gdwitt]

A controlled study of the effect of Tax Efficiency on actual Income for late game factions
I reached a level 8 imperium with Fame level somewhere above 300 in my latest game as WRE in Attila. I had 61-69 regions during this time.
I started to record tax and corruption values over 10 turns as I adjusted the tax tables in campaign_variables at least twice turn.
Method: Change stats. Launch. Click the treasure chest. Check 2 cities for corruption levels. The tax revenue changes at game load, not at the start of the next turn.
Uncontrolled variables included weather, bundles, battles, building completions, edicts and governors. Enough data did allow trends to emerge.
Variables considered: tax_efficiency_regions_maximum 100
tax_efficiency_regions_middle 40
tax_efficiency_regions_minimum 1
tax_efficiency_result_maximum_reduction 60
tax_efficiency_result_middle_reduction 40
tax_efficiency_result_minimum_reduction 5
and all “att_effect_economy_corruption_mod”s across Data and my mod totaled.
Conclusions:
Neither min_regions nor Tax_Efficiency_Result (TER)_minreduction seemed have any effect on tax level when a faction has over 60 territories.
Fig 1: TaxEfficRegionMax gives better returns when it is greater than 15 above one's own regions, but falls when it is more than 30 or so regions what your faction possesses. Once the value falls below the faction regions, there are no reductions in taxes.

Fig 2: TaxEfficRegionsMiddle: The closer this value is to your regions, the greater the tax return.

It seems that both variables "pull" at the taxes; the formula does not just look at TERegion value, but both. In other words, the TEReduction looks towards both TERegions - max and middle - but borrows more from the closest one.
For example, reducing the middle-tier tax regions will cause the TEReduction to choose a rate closer to the TERMaxRegion's rate. Taxes will increase.
The tax rates associated with these brackets.
Fig 3: TaxEfficResultMaxReduction: Inverse but steep slope. Each 1% increase corresponded to about a loss in about 1000 in taxes.

Fig 4: TaxEfficResultMiddleReduction: Inverse but shallow slope. Each 1% increase corresponded to about a loss of about 500 in taxes. This in effect is a lower tax rate.
Corruption value are directly restrained by Reductions. At corruption values under 95%, it seems each 1% increase in TERMaxReduct resulted in a 1% increase in corruptions values in each region. The relation with TERMiddlerReduc is less clear, but higher values corresponded with higher corruption.

Corruption-reducing buildings, edicts and traits will still increase income when the TEReduction, but there might be a threshold where this is limited. This is explored in the study below.
Income rises as corruption decreases. in an inverse relationship. For every increase of 20% corruption from effectbundles, taxes drop about 19k.
Fig 5: The slope of the line is about -0.0013. Corruption peaks out at a limit set by tax_efficiency_result_maximum_reduction.
In this setup, that upper limit was at 95 corruption and above. So keep in mind that the "70" doesn't correspond to the visible corruption drop.

Fig 6: Tracking the corruption by the city view has the same curve shape, but decreases more slowly. This is probably because tax rate also decreases with effect-level corruption and is not reflected in these city-corruption figures.
There appears to be base tax-reducing such as corruption that cause tax income to go negative and actually city corruption to be about 50 points above the corruption rates selected in effects.

Conclusions:
I think for all empires, vanilla values are pretty good.
Larger empires bringing in too much income would be better served with TERMaxReduction of around 70% and a TERMidReduction of about 35%. This assumes that trade income has been buffed and the empire has about 6 or so vassal states which seem to provide about 3500 income per turn.
I personally use 100,40,1,70,35,5 for Tax Efficiency values and rely on a trade-GDP at 2.1% and an AI system where it is fairly easy to hold onto 6-8 puppet states by imperium level 7 and 9 by imperium level 8. I have corruption set to 55% at Imperium level 8 and Tax mod at only 12% across all tables.
My income as WRE or ERE is still around 29000 per turn at turn 63. Why are 40 and 35% so low relative to the top tier? This is to make it easier for mid-level empires to progress.