In working on a future post I noticed my annoyance at Marx' use of the prime (${}^{\prime}$) symbol for the "rates" of various things, such as ${s}^{\prime}=s/v$ for the rate of exploitation. The mathematician in me doesn't like this since ${}^{\prime}$ is often used for derivatives. For example when dealing with rocket engines one often speaks of an engine's mass flow rate $\delta \phantom{\rule{0}{0ex}}m/\delta \phantom{\rule{0}{0ex}}t={m}^{\prime}=\stackrel{\xb7}{m}$ in units of kg/s. But since both $s$ and $v$ have the same units (or else we can't add them), Marx' ${s}^{\prime}$ is unitless, typically expressed as a percentage.

I am also annoyed with the word "rate" in the original German text of *Capital*, used verbatim in the English translation by Moore and Aveling.
Swedish translations tend to use the word "kvot" (en: *ratio* or *quotient*) instead.
Marx also frequently switches between "rate" and "grad" (sv: *grad*, en: *degree*) causing even more grief.
Clearly something needs to be done.

To get us started, what notation should we use for the derivative of the rate of exploitation? Perhaps $\delta \phantom{\rule{0}{0ex}}{s}^{\prime}/\delta \phantom{\rule{0}{0ex}}t$? We can't use ${s}^{\prime \prime}$ or ${\stackrel{\xb7}{s}}^{\prime}$ because that would be silly. One option is to first switch to the bar ($\overline{}$) symbol for Marxian ratios: $\overline{s}$. This allows us to use ${\overline{s}}^{\prime}$ or $\stackrel{\xb7}{\overline{s}}$ as shorthands for $\delta (s/v)/\delta \phantom{\rule{0}{0ex}}t$. The Newtonian $\stackrel{\xb7}{\overline{s}}$ has the benefit of not conflicting with Marx' notation.

$\stackrel{\xb7}{\overline{s}}$ has units of Hz (s^{-1}), but didactically it makes more sense to say %/s or %/year depending on scaling.
What of $c$, $v$ and $s$ themselves?
It has been pointed out to me by David Zachariah that this triad has units of *persons*.
This makes accumulated value have units of *person-seconds*:

${s}^{(-1)}=\int s\phantom{\rule{0.167em}{0ex}}d\phantom{\rule{0}{0ex}}t\phantom{\rule{0.333em}{0ex}}\text{[Ps]}$

where ${}^{(-1)}$ is used to denote an antiderivative.
A physical unit for people would be useful, and I have used *P* above.
It does have the downside of already being used for poise.

If we're already in the business of putting Marx' atrocious notation "back on its feet" then we could ditch Marx' $C=c+v$ and use the usual uppercase shorthands for antiderivatives:

- $C={c}^{(-1)}$ for historically accumulated and depreciated fixed and circulating capital (dead labour)
- $V={v}^{(-1)}$ for the present working population and its petty personal possessions (living labour)
- $S={s}^{(-1)}$ for historically accumulated and depreciated surplus (pyramids, yachts etc.)

All three of these are, of course, capital letters.

I have brought up depreciation here to highlight that $c$, $v$ and $s$ should themselves be broken down as follows:

$\begin{array}{rl}c& ={c}_{+}-{c}_{-}\\ v& ={v}_{+}-{v}_{-}\\ s& ={s}_{+}-{s}_{-}\end{array}$

The positives correspond to production and the negatives to depreciation. In a steady state economy production precisely equals depreciation. More fully we have this:

$\begin{array}{rl}C& =C(t)=\underset{-\mathrm{\infty}}{\overset{t}{\int}}c(\tau )d\phantom{\rule{0}{0ex}}\tau =\underset{-\mathrm{\infty}}{\overset{t}{\int}}({c}_{+}(\tau )-{c}_{-}(\tau ))d\phantom{\rule{0}{0ex}}\tau \\ V& =V(t)=\underset{-\mathrm{\infty}}{\overset{t}{\int}}v(\tau )d\phantom{\rule{0}{0ex}}\tau =\underset{-\mathrm{\infty}}{\overset{t}{\int}}({v}_{+}(\tau )-{v}_{-}(\tau ))d\phantom{\rule{0}{0ex}}\tau \\ S& =S(t)=\underset{-\mathrm{\infty}}{\overset{t}{\int}}s(\tau )d\phantom{\rule{0}{0ex}}\tau =\underset{-\mathrm{\infty}}{\overset{t}{\int}}({s}_{+}(\tau )-{s}_{-}(\tau ))d\phantom{\rule{0}{0ex}}\tau \end{array}$

Let us see what different sorts of "rates" we can reason about with this more sensible notation.
Rate of exploitation (*utsugningskvot*) and rate of profit (*profitkvot*) look just about the same:

$\begin{array}{rl}{s}^{\prime}& =\frac{S}{V}\\ {p}^{\prime}& =\frac{S}{C+V}\end{array}$

But we can also look at the situation like this:

$\begin{array}{rl}{\overline{s}}_{+}& =\frac{{s}_{+}}{{v}_{+}}\\ {\overline{p}}_{+}& =\frac{{s}_{+}}{{c}_{+}+{v}_{+}}\end{array}$

If we think about this very carefully we will realize that ${\overline{p}}_{+}$ here is close to the Shaikhian notion of *regulating capital* (${\pi}_{t}$), the "instantaneous" rate of profit.
This is *not* the same as the current historical profit rate ${p}^{\prime}$, nor will integrating Shaikhian regulating profit rates result in this historical profit rate:

$\begin{array}{rl}{p}^{\prime}=\frac{S}{C+V}& =\frac{\int s\phantom{\rule{0.167em}{0ex}}d\phantom{\rule{0}{0ex}}t}{\int (c+v)d\phantom{\rule{0}{0ex}}t}\\ {\overline{P}}_{+}={\overline{p}}_{+}^{(-1)}& =\int \frac{{s}_{+}}{{c}_{+}+{v}_{+}}\phantom{\rule{0.167em}{0ex}}d\phantom{\rule{0}{0ex}}t\\ \frac{\int s\phantom{\rule{0.167em}{0ex}}d\phantom{\rule{0}{0ex}}t}{\int (c+v)d\phantom{\rule{0}{0ex}}t}& \ne \int \frac{{s}_{+}}{{c}_{+}+{v}_{+}}\phantom{\rule{0.167em}{0ex}}d\phantom{\rule{0}{0ex}}t\end{array}$

This quirk is at the heart of an ongoing academic debate between Anwar Shaikh and Victor Magariño on one side and Paul Cockshott on the other. If however ${c}_{+}$, ${v}_{+}$ and ${s}_{+}$ are held constant then we expect $\underset{t\to \mathrm{\infty}}{lim}\overline{P}={p}^{\prime}$ since all capital and surplus eventually depreciates.

I will also introduce the following two definitions:

$\begin{array}{rl}{\overline{s}}_{-}& =\frac{{v}_{-}}{{s}_{-}}\\ {\overline{p}}_{-}& =\frac{{c}_{-}+{v}_{-}}{{s}_{-}}\end{array}$

Just as ${\overline{p}}_{+}$ regulates the rate of profit, so does ${\overline{p}}_{-}$.
We could therefore call ${\overline{p}}_{-}$ the *regulating depreciation*.
We have three constituent variables to consider here, with their own effects:

${c}_{-}$: the rate of depreciation of constant capital. The faster constant capital depreciates the higher the rate of profit, since capital does not have to maintain older less productive means of production. One reason war happens is because bombing factories increases ${c}_{-}$ which in turn increases ${\overline{p}}_{-}$ and over time ${p}^{\prime}$. This is detrimental to individual capitals, but beneficial to capital as a whole.

${v}_{-}$: the rate of depreciation of variable capital, or the rate of depreciation of labour power. One obvious way this happens is through death. It also explains why bourgeois parties are pro privatizing elderly care. Either you can afford it or you die, which means unproductive workers are disposed of more quickly. As a bonus, capital profits off of the privatized elderly care itself. Another way to raise ${v}_{-}$ is to increase the pension age. This forces workers to be productive for more of their lives, followed by hopefully dying as soon as possible after retirement. Yet another way to increase ${v}_{-}$ is to simply steal the workers' personal property. This can happen through legalized gambling, but also happens through regular theft, thievery being a fundamentally petty bourgeois occupation. Turning workers into slaves is yet another way to raise ${v}_{-}$, which happens today in places like Libya.

${s}_{-}$: the rate of depreciation of surplus value. The bourgeoisie wants ${s}_{-}$ to be kept as low as possible. They do this by investing their surplus into things that depreciate slowly, things like gold, silver, fine art, antiques and so on. The ultimate thing that never depreciates here on Earth is land and bodies of water. The latter is not yet privatized but bourgeois activists like Nalle Wahlroos, who has been advocating privatizing the Baltic Sea(!) and the atmosphere(!!), are hard at work on the issue.

Notice how war does not alter the rate of exploitation while it increases the rate of profit. This is one reason why it is in workers' interests to oppose bourgeois wars, beyond them causing death and suffering. War also involves the transfer of ownership of land. Because land doesn't depreciate this is another explanation for bourgeois war.

While land itself usually doesn't depreciate, what is on the land can and does depreciate. Forests for example are bookkept separately from the land they are on, and will depreciate if not looked after properly. One way land can depreciate is by being lost to rising sea levels.

How about the change in rate of exploitation over time? Because ${\overline{s}}_{+}$ is a ratio we must make use of both the product rule and the chain rule:

$\begin{array}{rl}{\stackrel{\xb7}{\overline{s}}}_{+}& =\frac{\delta}{\delta \phantom{\rule{0}{0ex}}t}\frac{{s}_{+}}{{v}_{+}}\\ & =\frac{1}{{v}_{+}}\frac{\delta \phantom{\rule{0}{0ex}}{s}_{+}}{\delta \phantom{\rule{0}{0ex}}t}-\frac{{s}_{+}}{{v}_{+}^{2}}\frac{\delta \phantom{\rule{0}{0ex}}{v}_{+}}{\delta \phantom{\rule{0}{0ex}}t}\\ & =\frac{{\stackrel{\xb7}{s}}_{+}}{{v}_{+}}-\frac{{s}_{+}\phantom{\rule{0}{0ex}}{\stackrel{\xb7}{v}}_{+}}{{v}_{+}^{2}}\end{array}$

The result has units of Hz. Informally we see the rate of exploitation goes up if profits increase, and down if wages increase. For the rate of profit the situation is similar:

$\begin{array}{rl}{\stackrel{\xb7}{\overline{p}}}_{+}& =\frac{\delta}{\delta \phantom{\rule{0}{0ex}}t}\frac{{s}_{+}}{{c}_{+}+{v}_{+}}\\ & =\frac{1}{{c}_{+}+{v}_{+}}\frac{\delta \phantom{\rule{0}{0ex}}{s}_{+}}{\delta \phantom{\rule{0}{0ex}}t}-\frac{{s}_{+}}{({c}_{+}+{v}_{+}{)}^{2}}\phantom{\rule{0.167em}{0ex}}\left(\frac{\delta \phantom{\rule{0}{0ex}}{c}_{+}}{\delta \phantom{\rule{0}{0ex}}t}+\frac{\delta \phantom{\rule{0}{0ex}}{v}_{+}}{\delta \phantom{\rule{0}{0ex}}t}\right)\\ & =\frac{{\stackrel{\xb7}{s}}_{+}}{{c}_{+}+{v}_{+}}-\frac{{s}_{+}({\stackrel{\xb7}{c}}_{+}+{\stackrel{\xb7}{v}}_{+})}{({c}_{+}+{v}_{+}{)}^{2}}\end{array}$

The above two formulas describe what we could call the *indirect* effects of changes in $c$, $v$ and $s$.
If we instead differentiate the two rates with respect to the three variables we get the following results:

$\begin{array}{rl}\frac{\delta \phantom{\rule{0}{0ex}}{\overline{p}}_{+}}{\delta \phantom{\rule{0}{0ex}}{c}_{+}}& =-\frac{{s}_{+}}{({c}_{+}+{v}_{+}{)}^{2}}\\ \frac{\delta \phantom{\rule{0}{0ex}}{\overline{p}}_{+}}{\delta \phantom{\rule{0}{0ex}}{v}_{+}}& =-\frac{{s}_{+}}{({c}_{+}+{v}_{+}{)}^{2}}\\ \frac{\delta \phantom{\rule{0}{0ex}}{\overline{p}}_{+}}{\delta \phantom{\rule{0}{0ex}}{s}_{+}}& =\frac{1}{{c}_{+}+{v}_{+}}\end{array}$

Note the similarity with ${\stackrel{\xb7}{\overline{p}}}_{+}$. The first two relations tell us that increases in the amount of value that has to be ploughed into inputs (${c}_{+}$) or wages (${v}_{+}$) affect the rate of profit negatively and in the same way. It makes no difference to the capitalist whether he (or she!) has to spend money on more coal or on hiring more people. The third tells us that an increase in the rate at which surplus is extracted raises the regulating profit rate.

$\begin{array}{rl}\frac{\delta \phantom{\rule{0}{0ex}}{\overline{s}}_{+}}{\delta \phantom{\rule{0}{0ex}}{c}_{+}}& =0\\ \frac{\delta \phantom{\rule{0}{0ex}}{\overline{s}}_{+}}{\delta \phantom{\rule{0}{0ex}}{v}_{+}}& =-\frac{{s}_{+}}{{v}_{+}^{2}}\\ \frac{\delta \phantom{\rule{0}{0ex}}{\overline{s}}_{+}}{\delta \phantom{\rule{0}{0ex}}{s}_{+}}& =\frac{1}{{v}_{+}}\end{array}$

Here the similarity is with ${\stackrel{\xb7}{\overline{s}}}_{+}$. We see first that it makes no difference to the workers how much dead labour is embodied in the inputs. They arrive "for free", their value simply being passed onto the output. What the workers do care about are the second and third relations. Higher pay -> less exploitation, more unpaid work -> more exploitation.

This same analysis could be carried out for ${\overline{s}}_{-}$ and ${\overline{p}}_{-}$ but this post is already long enough so I leave that as an exercise for you dear reader.