Part 34: Economic Breakdown 1

Empire size and food growth - What are my returns?

Welcome back to this bonus economic update for my Rise of the Samurai LP. In this update, I will be discussing some hard numbers behind a very common question that has been the cause of some debate amongst Shogun 2 players (and even in this thread!). While the numbers are only tedious to calculate, the many variables involved and the playstyles of certain players means that there is no magic bullet where one thing is always the right thing to do, despite what others might tell you.

This question is the ever popular - "Is it worth upgrading my farms if I don't need the food?"

Obviously, this is a question of an economic nature and so will need to be compared to the cost effectiveness of other economic improvement options, but those other options are outside the scope of this post. Instead, this post will present the results of multiple scenarios in which you would be considering upgrading your farms and present the results in a manner that will allow an informed comparison with your other options.

First, let us look at the farm chain of buildings:
The columns are Cost to build, town wealth gained, town growth gained and food gained respectively. Each province starts with a Fields and so can be ignored for the purpose of this document. All base wealth boosts are before fertility modifiers have been applied - Barren (1x), Meagre (1.3x), Average (1.5x), Fertile (2x), Very Fertile (3x).

Since we are looking at an upgrade comparison, we are interested in the difference between each building, so let's have a look at what each upgrade gives us in real terms:
Now that we have a cost for each upgrade and the benefits of each item, we need to show our methodology for how we are calculating the boost to our empires economy.

The amount of income added to your empire on a given turn from the upgrade is calculated as:
(Building Wealth bonus+(Building growth bonus x Turns since built)+(Food Bonus x Empire Size x Turns Since Built)) x Tax Rate

For the mathematically inclined, it would look like this:
(a+(b+(f*e))x)t

Where a is building wealth, b is building growth, f is the food bonus, e is empire size, t is the tax rate and x is the number of turns since construction.

Since each building adds only a single food and no additional town growth, we simplify it to:
(a+ex)t

Which can also be represented as:
at+etx

This is not the whole story however, as the tax rate can vary from province to province. Instead, we elect to have 2 tax rates, one for the local province and one for the empire. While all the local bonuses are affected by the (usually higher) local tax rate, your empire wide tax rate (which is what generates wealth from your empire growth) is likely to be significantly lower and is inversely impacted as your empire increases in size. Indeed, as your empire grows, your administrative cost will increase, reducing your empire tax rate from it's base of 30%.

We will call the empire tax rate m and now our formula looks like this:
at+emx

However, we are interested not in the income added on a single turn, but instead the total income that upgrade has given us up until our selected point. To find that, we must integrate our above formulae with respect to the turn.

This gives us the following result:
atx+(emx^2)/2 + c

Where c is a constant. As we will only be taking the integral between two values of x (0 and turns since upgrade), the value of c will cancel out. In addition, at x=0, the result is 0 and so we will just calculate for x is turns since upgrade to get our result.

We are interested in three separate "milestones" for our economy.

The first is when a building has paid for itself and so has become cash neutral. This is a useful metric to compare the break-even points of buildings, but is primarily of interest as a "tie-breaker" of sorts and is the minimum we expect of our buildings (as money now is worth more than money later, so at this point we have still made a loss in real terms).

The second is when a building has earned it's initial investment back, that is, it has earned enough to enable a second building to be constructed. This is a useful mid term indicator as it demonstrates the income growth pattern of a building. The closer this is to middle of our first and third indicators, the worse the overall growth potential (ie. close to a static return).

The third and final metric we use is when a building has earned three times it's initial cost. When combined with the other two data points allows us to see the trend for the building and this represents the long term returns of the building. It allows for more intelligent investment choices when you can weigh more money sooner against less money later or vice versa based on your situation (Small investments with good long term growth can be quite intelligent choices, while larger investments that only pay off in the long term are inherently more risky).

When we discuss risk in the context of Shogun 2 economic buildings, we are talking about things going wrong that prevent your investment from being the best return. These include invasion and loss of territory, enemy armies burning down farms on the campaign map and so incurring an associated (quite hefty) repair cost and also taking new territory that has better economic investment options. Buildings with fast break even points are inherently less risky than those with longer break-evens or high costs (and so tend to have worse long term returns to balance them, but not always!).

We will consider 5 different empire scenarios:
2 provinces - Your starting empire
5 provinces - Early game with a bit of steam
10 provinces - Solid expansion in late early game
15 provinces - Mid game and approaching Realm Divide
20 provinces - End Game (or midgame approaching Realm Divide for Long games, around the latest that food investment will yield reasonable dividends in time for it to matter.)

Additional scenarios are left as an exercise to the reader if they wish to calculate further options with the above methodology.

Now that we have all the theory out of the way, let's look at some numbers:

Time to Break Even - 2 provinces : Empire tax Rate : 30%
Time to Double - 2 provinces : Empire tax Rate : 30%
Time to Triple - 2 provinces : Empire tax Rate : 30%
Analysis: With almost no growth due to the low province count, we see almost linear returns on investment, as the vast majority of the wealth generated is from the static wealth boost from the farm itself. The only outliers are those which are by themselves particularly poor investment options (See Rice Paddies in Barren soil following a significantly less linear pattern 55 turns for the initial, 85 turns for the doubling and 109 turns for the tripling, a reduction of 56 turns over a linear result) which should not be considered in all but the most unusual situations given that the normal game is 100 turns long. The numbers here clearly indicate that food based growth is very poor for small empires, as we would expect. Overly aggressive farm upgrades in the early game yield particularly poor returns compared to the other options available.

Time to Break Even - 5 provinces : Empire tax Rate : 27%
Time to Double - 5 provinces : Empire tax Rate : 27%
Time to Triple - 5 provinces : Empire tax Rate : 27%
Analysis: The extra few provinces have made a difference of around 10% in reduction to triple wealth for the middle improvements (which you would most likely be looking at) but still has had minimal impact on the high value upgrades. Rice paddies and Improved Irrigation at this size, even for Fertile provinces are unlikely to generate maximum return compared to other economic buildings, although the value of Dry Field Agriculture only increases it's already incredibly competitive economic position, able to pay it's investment back in a mere ~6 turns, although with this level of aggressive repayment we see a reduction in time to triple of 2 turns compared to linear, a 12.5% reduction (Roughly in our 10% bracket we expect), although this is due to the low initial cost making the growth a more substantial portion of the economic value. Improvements above Dry Field Agriculture still yield dubious returns, as by the time they have got up a head of steam, the game is well and truly over. Of course, you won't spend the whole game at 5 provinces, but early on it's important to realise that higher level farms are not the best economic return, even out to midgame.

Time to Break Even - 10 provinces : Empire tax Rate : 24%
Time to Double - 10 provinces : Empire tax Rate : 24%
Time to Triple - 10 provinces : Empire tax Rate : 24%
Analysis: At 10 provinces, we start to see a serious compression of the longest time outliers towards the median value, which is also decreasing at a slower rate. In the more extreme cases (Barren fertility) we see a it taking twice as many turns to earn three times it's cost compared to it earning back it's base cost, a major increase on linear time. However, the more valuable upgrades (Fertile, Very Fertile) see closer to a 20% reduction on linear time for their earnings to be triple their cost. At this size, high value provinces (40-50%+ tax rate and Fertile/Very Fertile) become competitive with other economic upgrades and will enter their full utilisation before the game ends. However, it should be noted that these are still only viable in the top end provinces at this stage as the arts required and time necessary to get to 10 provinces and have a province in the position to benefit fully are likely to have consumed some available time. If, on the other hand, you are performing extremely well, it's likely that even though you can build this early, you are quite likely to win before this has a chance to fire up and so other, more short term investments may be advised.

Time to Break Even - 15 provinces : Empire tax Rate : 21%
Time to Double - 15 provinces : Empire tax Rate : 21%
Time to Triple - 15 provinces : Empire tax Rate : 21%
Analysis: At 15 provinces, farms are starting to represent good value, with Rice Paddies providing around the same boost to your economy in the mid term as a Market (although only in Very Fertile provinces), with Improved Irrigation offering slightly better returns and beating a Market at Fertile or Very Fertile. At this point you are critically close to Realm Divide, so further expansion is unlikely until you are ready to begin your final push. Dry Field Agriculture only gets better and should be built in every province regardless of fertility if you have not already! This is a critical point in your Shogun 2 game for farms, as you have enough time left to take advantage of the growth boost from heavy food investment, but are large enough that you see a reasonable return. Regardless, improving your farms into Rice Paddies and beyond are really only economically sensible in Fertile or better, Average if you have no other investment options.

Time to Break Even - 20 provinces : Empire tax Rate : 19%
Time to Double - 20 provinces : Empire tax Rate : 19%
Time to Triple - 20 provinces : Empire tax Rate : 19%
Analysis: At 20 provinces in size, you are very close to the finishing line on a normal game. While the extra provinces help make food more competitive in absolute terms, you are unlikely to see real returns (more than your initial investment) in all but the most valuable of provinces (Which by this stage should have been well and truly upgraded.). Obviously if you are playing the longer game, the balance has shifted even more towards the return of food from other economic buildings, but it is improving at an ever slower rate, as the additional growth is eaten by extra Administrative Cost and the lack of cheap and easy options (You must push through Rice Paddies to get to Improved Irrigation) and these are only compounded by the hefty cost of Improved Irrigation which leaves the growth benefits proportional to it's cost rather lacking compared to other upgrades.

Conclusion: Farm development is a viable cornerstone of a growth strategy, but caution has to be undertaken as it is a long term plan which, if over-invested in, can paradoxically leave you worse off in the long term compared to other, more traditional economic buildings. Careful consideration of your immediate and near future options should be undertaken before investing in Rice Paddies or Improved Irrigation. It is, however, almost always worth upgrading to Dry Field Agriculture regardless of soil fertility, as the low cost for the growth boost makes it one of the best economic investments under all situations and a very close second in those where it is not optimal.

I hope you enjoyed the breakdown on the economic returns to farm upgrades. The sister piece to this post, Economic Buildings and long term returns will be up next, to give you the other half of the economic investment equation and let you make intelligent, informed decisions about where to invest your hard earned (or possibly looted!) koku.