Upgrade Market to Rice Exchange? An objective analysis

[Maltz]

Let's crunch some numbers! But we need to write a good equation to represent the problem.

Effective Tax Rate "ET%", Number of Turns "T"

The value of commerce, just like farming, mining, and town activities, is subject to taxing. At a normal tax rate of 30%, we tax 30% of the sum of farming, commerce, mining, town. So +300 commerce transfers to 90 koku of income after 30% tax.

But that's only in theory. In reality, we have admin cost. Admin costs goes up with the number of provinces. At just 1 province, the admin cost is 0. At 50 provinces, the admin costs goes up to more than 60% of the tax rate (but does not affect bonus tax rates from characters, retainers, and arts).

In mid-game, when we set tax rate at 30%, after admin cost, we only receive about 20% of it (more than 20% for early game, less than 20% for late game). The missing 10% goes to the government workers who collect taxes and... have unnecessary "business trips". So +300 commerce only gives us 60 koku.

But when we install a secret police (a Metsuke) into a province, we increases the effective tax rate even ABOVE the apparent tax rate. A level-2 Metsuke specialized in town supervision can easily achieve 30% of effective tax rate or above, at a normal tax rate of 30%.

In our analysis, we be flexible and call effective tax rate "ET%". Our gain from upgrading to Rice Exchange is increased by 300 * ET% (per turn). Let the number of turns be "T". The total commerce-income from Rice Exchange is therefore 300 x ET% x T.

Prediction: So how does the effective tax rate affect our decision to whether to upgrade markets? The higher the effective tax rate, the MORE actual income we receive from the +commerce building. So it should be a good idea to upgrade markets in towns we station a good Metsuke.

Town Growth, Basic Tax Rate "BT%", Number of Provinces "P", and Number of Turns "T"

Food surplus is a very strange concept in STW2. One food surplus gives +1 town growth in ALL provinces, not just one. You'd think that the surplus will become insignificant if it is shared by many provinces. There is one more factor to consider, that is the REDUCTION of town growth from base tax rate.

Updated: The Reduction of Growth "RS"

The actual town growth is affected by the base tax rate. After collecting a few dozen provinces of data, I cannot figure out the exact equation to calculate it (there seems to be hidden factors), but the following equation provides a very good estimate:

Let S = Sum of Town Growth Factors

This includes:
Town Growth from Infrastructures (Market, Ports, Roads)
Town Growth from Food Surplus
Town Grwoth from Other Factors (event, etc.)

Let RS = the Reduction of Sum of Town Growth
Real Town Growth = (1-RS)

At 0% Base Tax Rate, RS = 0% S (full town growth)
At 10% Base Tax Rate, RS = 6% S
At 20% Base Tax Rate, RS = 14% S
At 30% Base Tax Rate, RS = 26% S
At 40% Base Tax Rate, RS = 55% S
At 50% Base Tax Rate, RS = 100% S

But at very low province count (such as just 1 province), the above equation does not hold. Rather, we have a fixed value of reduction.

At 0% Base Tax Rate, RS = 0
At 10% Base Tax Rate, RS = 0
At 20% Base Tax Rate, RS = 1
At 30% Base Tax Rate, RS = 2
At 40% Base Tax Rate, RS = 9
At 50% Base Tax Rate, RS = 16

But since most of the time our territory count is rather high, and our tax rate is set at 30%, we may as well use the general case above.

Now let's come back. If you have 100 units of town growth potential from food surplus and town buildings, but set a base tax rate (BT%) of 30%, you only receive 100 x (1-0.26) = 74 units of town growth.

OK! Back to Rice Exchange. Upgrading Markets to Rice Exchange costs 1 food surplus in all provinces. Let the number of provinces be "P". The effect of the food consumption of Rice Exchange (grains dropped on the floor?) is therefore -P. This food consumption is then transferred to town growth as -(1-RS)P

But Rice Exchange also provides its own town growth boost at +5. That +5 is reduced by base tax rate to 5(1-RS) as well. At 30% base tax rate, we get 5 x 0.74 = 3.7 units of town growth.

So upgrading to Rice Exchange's final effect on town growth is (1-RS)(5-P). At 30% base tax rate, it is 0.74 x (5-P).

Town growth works over time. For example, with a town growth of +7:

On turn 1, 7 town
On turn 2, 14 town
On turn 100, 700 town

The total town growth across all provinces is calculated as (let the number of turns be T):

(1-RS) x [(5-P) + (5-P) x T]/2 x T

But that's not our income yet. Town activity is also subject to the average effective tax across all provinces (ET-AVE%). This is different from the effective tax rate in the previous case, since we probably can't put the same Metsuke into all provinces. (But if we don't use any Metsuke, ET-AVE% is probably identical to ET%).

So our income (if negative, a loss of income) from upgrading the rice exchange over T turns is:

Extra Profit = -1500 + 300 x ET% x T + (1-RS) x [(5-P) + (5-P) x T]/2 x T x ET-AVE%