I've been playing around with the math in the first half of das kap vol 3, trying to verify some things. He essentially takes basic accounting principles and extrapolates it for the whole of industry. However, he also qualitatively describes phenomena with words, which I believe can be more concisely described through formula.

What I did was essentially put his words into math. It does require a bit of calculus. I'd appreciate it if I could receive some criticism for this.

The equation is as follows:

C = c + v + s

where
C is the revenue / amount of money to buy all the goods being sold,
c is the cost of materials, depreciation of equipment, utilities, etc
v is the total amount of wages needed to sustain labour (assuming workers are getting paid enough to live) s is the surplus value (net profit)

This is just validated because it's identical to basic accounting. It's just a balance sheet rearranged.

We don't necessarily take wages as the factor to look at, but rather wages with respect to the cost of goods.

So we need to introduce three more values:

w = v/t as the total dollars paid per hour of work

p = s/t as the total profit earned per hour of work, or economic productivity

u as the total units of product produced

The equation then becomes:

C/u = c/u + (w + p) * t/u

C/u = the price per unit (PPU)

We then take the derivative of PPU with respect to wages (w), and we get

d PPU/dw = t/u,

t/u = the total time over the total amount of units produced, or the rate of production.

That is, the time it takes to produce one unit is proportional to how affordable that thing is. So, if you want affordability and an affordable cost of living, then you need to be able make things fast. If you take the derivative of PPU with respect to p, then you get the same equation. You get more economic productivity the faster you make it.

Remember this, we will circle back to this concept.

If we rearrange the equation, then we get

p = C/t - c/t - w

Recall p is the total profit earned per hour of work. If sales remains the same, then profit maximization would be to suppress wages (w), or to cut the rate of spending (c/t). Now, if swap w and p, wage maximization means suppressing profit. Here we see competing interests between wage and productivity, which explains why they never increase at the same rate.

When the means of production is collectively owned, then p and w both belong to the worker, and there is no longer competing interests.

Now, the rate of profit is defined as the ratio of surplus value obtained for each input of labour (costs) and material input:

P' = s/(v+c)

With values substituted in:

P' = (p * t/u)/(w * t/u+c/u)

And you look at software, c/u is negligible, because it's just energy and electrons. if 'w' is pushed to 0, then P' goes to infinite. This is why the tech industry is traded at such extreme valuations, because they're able to get a massive return on investment.

Now, recall the term t/u, and how we need t/u to be low to have affordability. If t/u goes to zero, then P' also goes to zero. That is, affordability hinders capital accumulation, because each unit of investment generates less returns.