Might as well plug this here as well: https://www.haerdin.se/blog/2022/09/21/on-marxian-notation/
TL;DR: I am annoyed at Marx' atrocious mathematical notation and try to do something about it. I also notice depreciation is not as well studied as production.
That's interesting I never thought about the implication about the prime symbol also being used for derivatives. Personally, I'm not very familiar with math notation (if I want to know about derivatives I'll just plug stuff into wolfram alpha). I come at the variables more from the standpoint of an accountant, which is why I agree that depreciation is much more important than people often talk about. In fact, depreciation, paired with investment, is the whole key to understanding the tendency for the rate of profit to fall, as well as capitalist class struggle. I wrote about that a bit here with the secular rate of profit.
The Secular Rate of Profit - CASPER Forum
Recently, because the rate of exploitation and rate of investment have both stagnated, the ratio of investment to depreciation has converged to 1, the source of the stagnation of the rate of profit the past 40 years.
@casperadmin Ooh, that's really interesting. I think Farjoun, Machover and Zachariah come to a similar conclusion. Depreciation counteracts the tendency of the rate of profit to fall, at least to an extent. But, empirical data suggests it is still falling despite this. Question is if there's a lower bound, say 1%, since few capitalists would make investment they'd never live to see returns on, I think.
@casperadmin Ooh, that's really interesting. I think Farjoun, Machover and Zachariah come to a similar conclusion. Depreciation counteracts the tendency of the rate of profit to fall, at least to an extent. But, empirical data suggests it is still falling despite this. Question is if there's a lower bound, say 1%, since few capitalists would make investment they'd never live to see returns on, I think.
I use depreciation a bit differently than others. Because Marx says that constant capital passes on value as its used up, it makes sense to me that depreciation /is/ the measure of constant capital, and in that sense not a countertendency but the tendency itself as it applies to rising OCC.
As for how it applies to capitalist decision making, I created this index of real consumable income to the capitalist class which is their real after tax and investment income - this is the money which goes to reproduce the capitalist class, and regardless of the rate, if this amount decreases it must either decrease the income to particular individuals or make the capitalist class smaller. A rising OCC demands a higher share of profit going towards investment, and thus less income going towards consumption. Something I prove here Simulating the Rate of Profit - CASPER Forum
The index in question should be attached. We see how important the dips during the great depression and the 70s profitability crisis were, as well as how neoliberalism has vastly expanded the wealth of the capitalist class.
@casperadmin Right, I only write about the depreciation of the fixed part of constant capital in the post. Circulating capital depreciating when it is used as input seems a reasonable addition. It also matters how the depreciation is accounted for. While in accounting one often uses a flat depreciation rate, from the point of view of production a machine hasn't actually physically depreciated until it is finally put out of use, besides maintenance.
I had in mind to also bring up how value creation happens as a continuous process whereas exchange happens in discrete steps. That'd make the post much longer, but it's a point that makes my formulation slightly different from Shaikhs.
I had a realization earlier today that what I'm getting at might perhaps better be called "destruction" rather than depreciation, destruction then being the opposite of production. Then one could call pbar+ "regulating production" (roughly Shaikhian regulating capital) and pbar- "regulating destruction", and the symmetry between the terms is hopefully more easy to grasp. The productive process then destroys its inputs to create its outputs. Only when the machinery (fixed capital) is no longer useful is it actually destroyed, or rendered useless, regardless of whether this happens due to wear or moral depreciation.
Indirectly related: we now have a much better grasp of what calculus is *saying*. Specifically we can think of calculus as really discussing certain homomorphisms between rings of functions on smooth manifolds. From this things like the fundamental theorem of calculus follow naturally as well as intuitions as to what *concretely* is being said. I think while we are trying to inject rigor into Marx's math attention should be paid to what ring(s) we what to be discussing. LoC seems to be the starting point for this discussion.
I'm more familiar with the Math than the Econ. @thardin I'm interested in those subscript '-' and '+' you had, and their semantics. I can think of some sensible ones that fit with the foundations I am comfortable with.
@jules They are mostly arbitrary. I thought about "p" and "m" instead, or "prod" and "depr". Or given the discussion in here about depreciation really being destruction, maybe "prod" and "dest". I guess "d" covers both ways of looking at it, but we should be very clear what we mean. Whichever we pick we then get six symbols: c_p, v_p, s_p, c_d, v_d and s_d.
By the way, the reason I invert \bar{p}_- is so that it too is likely something that capital wants maximized. But we could also put the more expected definition, let's call it \bar{p}_d = s_d/(c_d + v_d), that capital instead seeks to push as close to zero as possible. Minimize s_d, maximize c_d and v_d. And of course from Marx we already have maximize s_p, minimize c_p and v_p.
At the bottom of the post you have ((partial s+)/(partial s+))=(1/v+); should that be((partial s+)/(partial p+))=(1/v+)?
At the bottom of the post you have ((partial s+)/(partial s+))=(1/v+); should that be((partial s+)/(partial p+))=(1/v+)?
Nah, the numerator is \delta \bar{s}_+ not \delta s_+ 🙂 In Marxian terms that would be ds' /ds which is why I dislike the s' notation since one might think s' itself is a derivative.
Perhaps a symbol other than bar should be used? I was also contemplating ^ (\hat) which would stand out more. Or just different symbols. IIRC Marx always uses p' and never p, unlike s' which competes with s for meaning. Instead one could maybe take p = s/(c+v) and e = s/v, e being short for exploitation. Both p and e then are "rates"
@thardin ahh I see, looking back at your post I can see the bar. In my work we use Latin alphabet when the we want to modify an existing model with new terms, so maybe that could work?
@madredalchemist How exactly do you mean? Just picking unused letters like I did with e above? Got any suggestions?
apologies I meant greek alphabet but it's often a good idea to use the equivalent in greek or something that looks similar but distinct i.e for s use sigma but if you have tuples of variables i.e (s,v,p) you could have (alpha, beta, gamma) etc. honestly you could really chose anything so as long as it stands out.
@madredalchemist Yeah, that's also one approach I had in mind. Say ρ and σ for p' and s', the integrals of which would be P and Σ. In a future post I currently use q and χ for two new concepts, and I don't want there to be contradictory notation.