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The Beaker Productivity of Banks

So, as promised in discussions for Adventure 38, I'm here to analyze yet another economic effect in Civilization 4.

City buildings that multiply cash production have a self-defeating behavior. The cash they produce enables the gold slider to be lowered, which provides less baseline cash for the buildings to multiply. Does this effect exist, and if so, how can we quantify it?

I make a couple simplifying abstractions again for this discussion. I assume that the research and cash sliders operate in increments of arbitrary precision rather than 10% (see my other discussion for more on that.) And I'm going to assume that cash serves no other purpose than paying city expenses towards research, so the only goal is beaker production. This is mostly true in most real games. Unit upgrades and cash rushing both seem worthwhile only in situations of emergency defense or end-game military mop-up once research is no longer a factor. The espionage and culture sliders are used only in specialized game situations.

Let's consider a simple quantitative example. Suppose we have a single city empire with the following characteristics.

Commerce per turn100
Expenses per turn35
Cash multiplier+25% (marketplace)
Science multiplier+50% (academy)

Therefore, this city is running the slider at 28% gold and 72% science, with the following outputs:

Gold28 commerce x 1.2535 (equal to expenses)
Beakers72 commerce x 1.50108

Building a bank looks like it should add 28 x 50% = 14 gold. But that very 14 gold allows us to lower the cash slider, making the bank itself worth less than 14 gold. The city's new outputs with a bank are these:

Gold20 commerce x 1.7535 (equal to expenses)
Beakers80 commerce x 1.50120

Because now we are running only 20% cash slider, the bank is worth only 20 x 50% = 10 gold per turn. The bank has defeated itself in the amount of 4 gold per turn. The effect is real.

But in another sense, this bank is producing no gold at all. Our gold production pre-bank and post-bank is exactly the same at 35. So what is this bank doing? It is producing research: 12 beakers per turn. Do note that this is less beakers than the bank's original apparent face value of 14 gold.

How did this happen? The bank "liberates" 8 commerce from the cash slider that can now go to the science slider instead, so it's 8 + 50% = 12 beakers. Stated generally, the beaker value of a bank is equal to the freed commerce times the science multiplier. What matters is not the direct cash productivity of the bank; what matters is the commerce freed up.

So far, I have calculated the effect of a bank only on commerce. Now let's consider economy created through the bank multiplying any direct gold production such as shrines and merchants. This gold is not self-defeating: the presence of the bank does not reduce its own baseline of direct gold - but does further reduce the commerce baseline input to the bank. That last effect is important but easily intuitive in game play anyway; direct gold production usually comes in quantities (shrines) that make a bank obviously high priority.

Thus the formula for the beaker productivity of a bank or grocer or market is this. The blue part up to the multiplication sign is the freed commerce. (For the moment again I'm ignoring the effect of direct gold defeating slider commerce.)

(Original commerce allocated to gold - New commerce allocated to gold) * Science multiplier

Since commerce allocated to gold, times the cash multiplier, equals expenses (by my original axiom that all cash production only pays maintenance), substituting gives us the following.

((Expenses / Old cash multiplier) - (Expenses / New cash multiplier)) * Science multiplier

In our example, ((35 / 1.25) - (35 / 1.75)) * 1.50 = 8 * 1.50 = 12. That example is for one city, although all the numbers are civ-wide. The old and new cash multipliers are the aggregate weighted average cash multiplier in all cities before and after adding the bank. The science multiplier is also civ-wide, because the freed commerce is spread across every city in converting to beakers.

Now look at what doesn't appear in the formula. Your sliders! A bank produces equivalent beakers regardless of your gold slider? That can't be right. Think of a compact empire running 98% science slider. A bank might bump that to 99% science which won't add many beakers.

The answer is that the gold slider is almost there but disguised. Expenses divided by the cash multiplier is the gold portion of the commerce slider. But what matters is the quantity of commerce that must be devoted to expenses, not the proportion of that commerce.

Here's proof. Suppose my example city goes up to 200 commerce while keeping the same 35 expenses. You might think the bank's productivity should double too, but it stays exactly the same. 35 expenses are still covered by exactly 1.25 * 28 commerce pre-bank or 1.75 * 20 commerce post-bank, so the bank still saves exactly 8 commerce which makes 12 beakers. The bank's beaker productivity is completely independent of the cash slider position. The extra 100 commerce goes entirely into science ignoring the bank.

This bears repeating: bank beaker productivity is independent of how much commerce you are producing. Consider civilization A whose expenses cost 80 commerce out of 100 total, versus civilization B whose expenses cost 80 commerce out of 500. Both will get the same beaker count from a bank.

There exists a cap on the productivity of a bank, thanks to the self-defeating effect. A bank cannot pay out any more beakers than you have expenses because the bank reduces its own baseline until that is true. Strictly speaking, the cap on bank beakers equals one-third of your expenses times your national beaker multiplier. The former is because the bank can't represent more than one-third of your cash production, because it adds 50% to the baseline 100%. If grocer and market are also present, it's one-fourth, adding 50% to prior 150%.

In fact, leaping even further on this train of thought, a bank functions more like cutting expenses than producing economy. Just like a courthouse says "-50% city maintenance", a bank really is "-33% commerce needed for expenses." And just like a courthouse, the saved commerce goes to science instead and multiplies by the national science multiplier. That -33% assumes no other cash multipliers and declines slightly with them; in my example it was 8/28 = -28.5% thanks to the existing market.

This effect is a significant part of why large empires always eventually outresearch smaller empires. Small empires with low expenses get next to nothing out of banks. Large empires essentially get an additional set of expense-reducing buildings that small empires do not. And smaller empires can have a greater proportion of their expenses paid by direct gold such as shrines and settled prophets, leaving even less expense headroom for the bank to work in. A bank can even break down completely; a small empire researching at 100% slider with surplus gold (negative net expenses) leaves the bank completely worthless for beakers.

Now, civilization B a few paragraphs up will get the same payout as civ A from banks, but much better payout from universities, which is my next topic. The same principle extends to grocers and markets which also produce cash, although the need for them is typically driven more by the health and happy caps. So I will continue to focus on banks specifically.

With the beaker output of a bank quantified, we can directly compare it to an observatory or university. In my example, 12 beakers from the bank is clearly less than the 72 x 25% = 18 beakers from either of the research buildings. This is intuitively correct; at 72% science slider we would naturally build research buildings first.

But enough theory, where's the practical upshot? So I have an empire that overexpanded into economic recession and is now recovering. When do I change priority from banks to universities and observatories instead? Intuitively, most of us would say over 50% science slider, or roughly there based on payout value factoring the lower hammer cost of the observatory and greater multiplier from the bank.

But the real determinant is not the slider, but rather your total expenses and your already-existing science multiplier buildings. Libraries make banks better because the saved commerce no longer paying expenses instead multiplies into more beakers. Corporate expenses make banks better by providing more expenses to pay before self-defeat. Courthouses make banks worse for the opposite reason. All these factors are independent of the slider position.

The breakover point can be algebraically calculated, although there's probably too many terms here for any real practical use.

BankBeakers = ((Expenses / OldCashMult) - (Expenses / NewCashMult) + (Commerce * ((NewCashSlider / NewCashMult) - (OldCashSlider / OldCashMult)) + DirectGold)) * SciMult

UniversityBeakers or ObservatoryBeakers = ((Commerce * ScienceSlider) + DirectScience) * 25%

If BankBeakers > UniversityBeakers, build the bank first. If BankBeakers > ObservatoryBeakers * 200/150, build the bank first, with that last term accounting for the hammer difference.

The green portion of the BankBeakers equation is the commerce saved by the bank and redirected to science. The blue portion (starting with "Commerce") is the real headache. This accounts for the bank multiplying direct gold production such as merchants, which is not self-defeating, but does defeat the portion of the bank's input that comes from the commerce slider. And "NewCashMult" and "NewCashSlider" in there are not trivial to figure; those are both nationwide values towards which this one bank contributes as part of a weighted average.

Even I am not going to work through that many numbers for each city. smile.gif - 1kb Bank versus university is intuitive in most cases just by looking at a city's science and cash production numbers. And when it's close enough that intuition may not suffice, the difference will amount to pennies, and also you'll just build both buildings in succession anyway.

Next, let me address the relative utility of a bank versus Wealth. In end-game situations, there may be little enough time until military victory or completing the spaceship research that the bank might not have enough time to pay off over Wealth. A bank produces no net value until it has generated 200 additional gold, because otherwise you could come out ahead by building Wealth instead. This is independent of hammer multipliers; +25% from a forge applies to both constructing the bank and to Wealth itself. The exception is Organized Religion, which applies only to constructing the bank, and lowers the breakpoint to 200 / 1.25 = 160 gold.

The self-defeating effect comes into play here as well. It is quantified the same as above except without the final multiplication through the science factor into beakers. My example city seemed to have the bank producing 28 x 50% = 14 gold per turn, but after the self-defeat of the commerce slider, it really produces 20 x 50% = 10 gold. So the bank payback time over building Wealth is not 200 / 14 = 14.3 turns, but 200 / 10 = 20 turns.

Finally, let me address how the "binary science" tactic figures into this. It doesn't. The productivity of a bank depends not on your slider, but on your expenses, which cannot be flipped around.

It's true that the zero phase of binary science enhances the bank, while the research phase renders it idle. But aggregated over time, these sum to the same total. Again consider my 100-commerce city with 35 expenses, paid by running the cash slider at 20% because 20 x 1.75 = 35. This city could run 80% science all the time, or it could run 0% science for two turns and 100% science for eight turns. Either way will give the same beakers. 80 x 1.5 x 10 turns = 100 x 1.5 x 8 turns.

Binary science does not affect the bank-versus-university decision, but does come into play if the decision comes out in favor of a science building. Then the correct play is to run zero science until the university is built, and then change to full research, so that the commerce-to-beaker conversion enjoys the greater science multipllier.

Now you pass Civ Economics 102. biggrin.gif - 1kb