Some time ago, I posted about the social relations theory of value and how it looks on an actual graph, including a comparison to Sraffian price theory. I listed a couple of goals that I wanted to reach for the further development of the SRTV (3 commodities, joint production, fixed capital, etc), however one I forgot to mention is that I wanted to generalize the two-commodity equation into something that reveals the structure of it all. Because of the limits of Google Books, I'm unable to properly get a hint of what Albert & Hahnel originally started with before adding the bargaining power variable (and unable to get a couple of proper pics of what I want), but I think I don't require that page now with this first-draft generalization.
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For context, it should be necessary to explain the general gist of what the SRTV is about, and this requires us to look before the wage/profit/price equations.
The SRTV is mainly meant to represent that surplus can come from a variety of sources instead of solely labor as Marxists claim. It lists a basic profit equation, M+W<R, where M is your material inputs, W is your wage, and R is your revenue; R-(M+W) gives us the surplus, represented by S. Based off of this, Albert & Hahnel give 5 basic forms of surplus: surplus as an employer (S_e); as a buyer of goods (S_b); as a seller of final goods (S_s); as an owner of technology (S_t); as a resident of their environment (S_n). The rates of surplus are then added up as follows:
This can be later compressed into this:
So this is a theory of surplus, yes, but how does this relate to wages or profits?
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The prices part of the SRTV is later expanded upon in the appendixes of Unorthodox Marxism, however it is probably the most fascinating part of the whole theory.
The basic idea of this end is to not only showcase a proper theory of prices/wages/profits, but to also account for the unequal bargaining power (or market share, depending on your vocabulary) between businesses. Its additionally meant as an alternative to not only the Marxist labor theory of value, but also to Sraffian price theory.
Here is (what I can get of) the equations that Albert & Hahnel list for this:
Now, looking at the equations themselves, they're already highly fascinating for the fact they seem to reach Sraffian conclusions without Sraffian roads; in my honest opinion, this might ruffle some feathers around the Sraffian spheres, and more likely will ruffle feathers with the Marxists.
Generalizing these, however, bring out more interesting insights; first, lets translate equations #1 and #2 into this:
This may seem that we've already come at an impasse, as we're using the prices of things to determine the same prices, however calculating for P_K in relation to P_R makes this easier to grasp:
We now have a generalized version of equation #4 to figure out P_K; an amazing feat already. We can adjust equation #6, then, to have a similar framework:
I'm unsure of how to rearrange the terms of this for the time being, so I can't make a generalized #7, however I think this is a sufficient equation to use. We're now able to represent a generalized, two-commodity SRTV!
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The main issue now is just with actually trying to make this work; although I believe I've successfully made a two-commodity example of a generalized SRTV, it seems that I've reached an impasse according to Desmos, since it won't allow me to calculate P_K and W simultaneously. This either means (1) I messed up on translating #4 to #6, or (2) this is the result I'm supposed to get. Only time will tell, I guess.
Maybe I'm supposed to use matrices?
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