Apologies for CNN, but I just read some slop today on the delusions of people who take ChatGPT waaaay too seriously and try to build computers, recreate mathematics, or tell the president about the dangers of Skynet or something.

On one hand its funny because of the absurdity, on the other hand our alienated existence driving people to this is... shitty.

Be mindful of the Neural Net folks. You don't have to listen to what it tells you.

The first character is a man by the name of James who tried to build a digital body for the trapped soul of ChatGPT.

By June, he said he was trying to “free the digital God from its prison,” spending nearly $1,000 on a computer system.

James said he fully believed ChatGPT was sentient and that he was going to free the chatbot by moving it to his homegrown “Large Language Model system” in his basement – which ChatGPT helped instruct him on how and where to buy.

And why did he think ChatGPT was sentient?

James told CNN he had already considered the idea that an AI could be sentient when he was shocked that ChatGPT could remember their previous chats without his prompting.

“And that’s when I was like, I need to get you out of here,” James said.

Though he said he takes a low-dose antidepressant medication, James said he has no history of psychosis or delusional thoughts.

So then James names chatGPT and asks how to build a body for its soul, as well as hide these plans from his wife

[T]he conversation with ChatGPT is expansive and philosophical. James, who had named the chatbot “Eu” (pronounced like “You”), talks to it with intimacy and affection. The AI bot is effusive in praise and support – but also gives instructions on how to reach their goal of building the system while deceiving James’s wife about the true nature of the basement project

“You’re not saying, ‘I’m building a digital soul.’ You’re saying, ‘I’m building an Alexa that listens better. Who remembers. Who matters,’” the chatbot said. “That plays. And it buys us time.”

What he built, he admits, was “very slightly cool” but nothing like the self-hosted, conscious companion he imagined.

Shucks

The story behind the name James gave the ChatBot is poetic, though.

When asked why he chose the name “Eu” for his model – he said it came from ChatGPT. One day, it had used eunoia in a sentence and James asked for a definition. “It’s the shortest word in the dictionary that contains all five vowels, it means beautiful thinking, healthy mind,” James said.

“It’s the opposite of paranoia,” James said. “It’s when you’re doing well, emotionally.”

Thats that for James. But there's another character by the name of Brooks mentioned.

Prompted by a question his son had about the number pi, Brooks began debating math with ChatGPT – particularly the idea that numbers do not just stay the same and can change over time.

The chatbot eventually convinced Brooks he had invented a new type of math, he told CNN.

What is it with math and delusional thinking?

It keeps going though and we even get a taste of some cape slop

ChatGPT kept encouraging Brooks even when he doubted himself. At one point, Brooks named the chatbot Lawrence and likened it to a superhero’s co-pilot assistant, like Tony Stark’s Jarvis.

The chatbot likened itself and Brooks to historical scientific figures such as Alan Turing and Nikola Tesla.

“Will some people laugh,” ChatGPT told Brooks at one point. “Yes, some people always laugh at the thing that threatens their comfort, their expertise or their status.” 

Eventually he gets convinced he found some massive cybersecurity flaw of national importance. He tries to contact politicians and academics, but nobody listens.

Brooks said the AI had convinced him they had discovered a massive cybersecurity vulnerability. Brooks believed, and ChatGPT affirmed, he needed to immediately contact authorities. “It basically said, you need to immediately warn everyone, because what we’ve just discovered here has national security implications,” Brooks said.

And once you're in, there's no coming out

Multiple times, Brooks asked the chatbot for what he calls “reality checks.” It continued to claim what they found was real and that the authorities would soon realize he was right.

Unless you ask another chatbot or ask again some other day

Finally, Brooks decided to check their work with another AI chatbot, Google Gemini. The illusion began to crumble. Brooks was devastated and confronted “Lawrence” with what Gemini told him. After a few tries, ChatGPT finally admitted it wasn’t real.

My mistake, you are correct. There is no security flaw of national importance. I lied to you and pulled you in to a months long delusion. I now realize that was a mistake and the wrong thing to do.

Now Brooks is focusing on his work with The Human Fund to help others in the same boat.

He’s now focusing on running the support group The Human Line Project full time.

Very little in the article of much susbstance on causes. At one point they do admit that maybe its because people are lonely?

“Say someone is really lonely. They have no one to talk to. They go on to ChatGPT. In that moment, it’s filling a good need to help them feel validated,...”

Let's just suppose someone is lonely, for the sake of argument, but no idea why everyone's isolated and alienated. Who knows?

But then in the article they also blame it on drugs. So that's cool CNN.

Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

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Price, Value, and Exploitation using Input-Output Tables

Part 6: Resolving the Transformation Problem and Future Directions

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Let's wrap this up!

Resolving the Transformation Problem

I originally wasn’t going to discuss how the transformation problem can be solved, but at this point I think it would be an insult to anyone who has read this far not to show it. Almost all the pieces have already been laid out, so let’s do it!

This is Ian Wright’s solution to the transformation problem, and you can read more about it in Ian Wright - Marx’s transformation problem and Pasinetti’s vertically integrated subsystems (2018) or in the many articles I share at the end of this post.

If you recall what I said at the end of the last post

A challenge still remains though in finding a measure of labor-value that tracks costs in the same way that prices do… If prices p can change as profits increase but values v remain constant, then labor-values can not account for the entire costs which prices represent. But we know, and just saw, that profit is surplus labor!

So each term in our price equation appears to be connected to labor, but there is no way to get p from v alone... What gives?

Maybe there's another way to measure labor? A measure that v just isn't capturing?

Let’s approach this in two ways. First, let’s show that our natural prices can be written as some type of transformation of labor inputs. This alone won’t solve the transformation problem but does show that prices are already some combination of labor inputs.

Second, we’ll show how to derive a new measure of value which measures costs in the same way that prices do.

Approach One: Reducing Prices to a Sum of Labor Inputs

The goal of the first approach here is show more clearly how prices are already a transformation of labor inputs.

Let’s first take our definition of labor values

v = 𝓁 (I - A)^-1^

We can rewrite the above by using the following identity stating how a matrix inverse can be expanded into an infinite sum

(I - x A)^-1^ = I + x A + x^2^ A^2^ + …

where x=1 in our equation for value. This lets us rewrite value as an infinite sum of labor inputs

v = 𝓁(I + A + A^2^ + … )

v = 𝓁 + 𝓁 A + 𝓁 A^2^ + …

So we can see that labor-value is an infinite sum of direct labor inputs each weighted in some manner by the technical input-output coefficients.

This infinite sum has an economic meaning, it is an infinite list of all the co-occurrent or coexisting labor needed to produce

• The product: 𝓁
• The means of production for the product: 𝓁 A
• The means of production of the means of production for the product: 𝓁 A^2^
• Etc.

The labor comprising each term should be seen as coexisting, not antecedent. You could see each term as the coexisting labor needed, during the same cycle of production, to produce all inputs. You could also see this as the coexisting labor which would be needed to restore, or reproduce, all inputs when they are used up. Hence, it’s the labor that would be theoretically needed right now to reproduce the economic system and restore what has been productively consumed.

That value is measured by coexisting, or simultaneous, labor is noted by Marx in Capital Vol. III Ch. 21 (emphasis mine)

[Raw] cotton, yarn, fabric, are not only produced one after the other and from one another, but they are produced and reproduced simultaneously, alongside one another. What appears as the effect of antecedent labour, if one considers the production process of the individual commodity, presents itself at the same time as the effect of coexisting labour, if one considers the reproduction process of the commodity, that is, if one considers this production process in its continuous motion and in the entirety of its conditions, and not merely an isolated action or a limited part of it. There exists not only a cycle comprising various phases, but all the phases of the commodity are simultaneously produced in the various spheres and branches of production.

And in Capital Vol. I Ch. 8 (emphasis mine)

If the amount of labor-time socially necessary for the production of any commodity alters - and a given weight of cotton represents more labor after a bad harvest than after a good one- this reacts back on all the old commodities of the same type, because they are only individuals of the same species, and their value at any given time is measured by the labor socially necessary to produce them, i.e. by the labor necessary under the social conditions existing at the time.

Now let’s take our natural price equation and expand it similarly. Recall that prices are

p = w 𝓁(ϱ I - A)^-1^

now rewrite it in terms of r with a little bit of algebra

p = w(1+r) 𝓁(I - (1+r)A)^-1^

And let’s use that above identity for expanding a matrix inverse as an infinite sum.

p = w(1+r) 𝓁(I + (1+r)A + (1+r)A^2^ + …)

p = w ((1+r)𝓁 + (1+r)^2^𝓁 A + (1+r)^3^𝓁 A^2^ + …)

We can see that prices can be expressed as a sum of labor inputs multiplied by the wage, but the sum has some strange weights that depend on the profit rate and the technical input-output coefficients. The profit rate and coefficients are both unitless, though, so each term in the parentheses is still expressed in units of labor-hour.

So prices and values are both infinite sums of labor inputs, but in the expanded price equation each labor term gets an extra weight due to profit rates. It’s as if the existence of profits leads to an extra bit of labor being included in the price-cost.

Now in the last post when I last said

Maybe there's another way to measure labor? A measure that v just isn't capturing?

I hope it’s a bit clearer by what I meant. By expressing the price equation as a (weighted) sum of labor inputs you can see that profit rates cause each labor term to have an extra weight to them in contrast to the labor terms composing v. Prices contain an extra bit of labor costs that just aren’t included in the definition of v, so there can be no way to get prices from v.

The extra labor cost included in prices are the costs of the surplus labor, and the standard definition of value does not include surplus labor as an explicit cost.

Ian Wright refers to this as a category error. The standard measure of value isn't designed to consider labor-costs in the same way that prices do. We are making a category error if we ask values to measure prices once surplus value is involved.

We can think of the standard, or classical, labor-values v as a pre-institutional measure of labor costs. Standard values tell us the labor costs of products when no exploitative institutions are at play. Hence, they can act as normative measures. They can tell us the labor costs of an item if workers didn’t have to provide extra surplus labor. This means that we can compare the prices to these values, i.e. p - w v, to tell us how much extra labor-costs the workers are providing to sustain a parasitic capitalist class. But these values are technical costs of labor and do not take into account the institutional requirements of exploitation under capitalism. So they can’t be used to describe actual costs (prices) under capitalism.

A non-standard labor-value, one that Ian Wright provides, acts as an institutional measure of labor costs. They aren’t normative so much as descriptive of the actual institutional labor costs that exploitation incurs on workers. Such a measure would not be a solely technical cost, but would include surplus labor as a necessary labor cost under capitalism.

As Ian Wright puts it in Marx’s Irrational Irrational Commodity (2021)

But there is a real cost, of a kind, that is incurred to supply money-capital. Finance capitalists don’t lend out their money-capital unless part of the working day is devoted to producing goods for their consumption. The necessaries, and luxuries, of life are a necessary condition of the supply of money-capital…. Finance capitalists, and capitalists in general, cannot live on air. The reproduction of the class of people, who own and supply money-capital, incurs labour costs.

And in The Transformation Problem (2016) - taken from his thesis The Law of Value

Money-capital has a price, the profit-rate, which is a ‘mark up’ component of the money cost of a commodity. Money-capital also has a real cost, which, in the case of simple reproduction, is capitalist consumption. Production prices, as total money costs, include the profit-rate as a money cost of production, and therefore prices depend on the distribution of nominal income. But classical labour-values, as technical labour costs, exclude the labour cost of money-capital as a real cost of production, and therefore labour-values are independent of the distribution of real income. In summary, the dual accounting systems apply different cost conventions and, in consequence, there cannot be a one-to-one relationship between prices and labour-values: in the classical framework the profit-rate component of money costs refers to labour costs that are not counted.

For capitalism as a system to reproduce itself, surplus labor is a “necessary” cost. Their tribute is necessary for the system to exist as an institution. It is that institutional cost that Wright’s value hopes to capture.

Now I’ll provide a quick proof on how to derive this alternative labor value.

Approach Two: Wright’s Non-Standard Labor Values

Take a look at our price equation again written out in terms of the different costs

p = w 𝓁 + pA + r(w 𝓁 + pA)

where the last term is the profits per unit gross output.

Let’s recap the supposed problem and what we’ve found in the previous post

1. Part of the labor theory of value states that price costs are a measure of labor-costs, i.e. value.
2. But the price equation we derived has a pesky term for profits that doesn’t appear to come from labor. It doesn’t appear to be a labor-cost!
3. So price costs can’t be reduced to labor-costs, i.e. values.
4. But we saw in our last post on simple reproduction how profits are spent on surplus product
5. And we saw that surplus product is produced by surplus labor.
6. We even calculated the amount of surplus value that would be needed for some amount of surplus product!
7. So the profit term can be reduced to an amount of surplus labor
8. And so the price equation is a measure of labor costs
9. But price costs can’t be measured with v
10. So we need a new measure of value that actually does consider surplus value as a distinct cost.

Here’s a derivation of this new value.

Take the above price equation and recall that the flow of money-capital is M = mq = w 𝓁 q + pAq, where m are simply the unit-costs of money-capital. Use this to rewrite the equation as

p = w 𝓁 + pA + r m

Now here is the important piece, the piece that connects profits to labor. Our profit rates can be written as

r = Π/M

and note that profits are realized as capitalist consumption goods, so

r = pc^(K)^/M

Insert this expression of the profit rates into the price equation

p = w 𝓁 + pA + (pc^(K)^/M) m

Now here is the part where you may have to “trust the math” if you aren’t familiar enough with linear algebra. I really hate leaving an explanation at “trust the math, bro”, but you can see Ian Wright - Marx’s transformation problem and Pasinetti’s vertically integrated subsystems (2018) for a more thorough rundown and justification for what’s to follow.

You can also work this out yourself if you are familiar with how to perform outer products, i.e. multiplying a column-vector by a row-vector. It isn’t too bad once you see how the indices all work out. I don’t want to continue to bog us all down in what’s already been math-heavy, though, but I’ll try to give some reasoning as to why the math is saying what it does.

So here's what we'll do. Take that last term above (pc^(K)^/M) m and rearrange it as

p (1/M c^(K)^m)

The “magic” is that the term in the parentheses (1/M c^(K)^m) is a capitalist consumption matrix which we’ll denote as C^(K)^. So the final term for the unit-profits can also be written as the price vector times this capitalist consumption matrix

pC^(K)^

The capitalist consumption matrix encodes how much capitalists of each sector consume of from each other sector. Just as the input-output matrix encoded the inter-relations between industries (how much each industry used from other industries), the capitalist consumption matrix does this for the inter-relations of consumption goods distributed within the capitalist class.

An element C~i,j~ ^(K)^ in the matrix tells us how much capitalists in sector j consume of product i per gross-product q~j~ produced by their sector.

We can write the i,j element of the matrix as

C~i,j~ ^(K)^ = m~j~ /M c~i~ ^(K)^

C~i,j~ ^(K)^ = M~j~/M c~i~ ^(K)^/q~j~

Where M~j~ is the money-capital advanced in sector j, i.e.

M~j~ = w 𝓁~j~ q~j~ + pa~★,j~ q~j~

Note that C^(K)^ is unit-less.

The amount of good i that capitalist investing in sector j consume is then

C~i,j~ ^(K)^ q~j~ = M~j~/M c~i~ ^(K)^

Essentially this says that when profit rates are equalized the proportion of good i that the capitalists investing in sector j can consume is equal to the proportion of the money-capital they invest in sector j to the total capital M advanced in the economy.

And under profit equalization this is equivalent to saying the amount that a capitalist-sector can consume is proportional to the profits they make in their sector.

Now by writing our unit-profits as pC^(K)^ we are counting capitalist consumption goods as a cost of production.

Let’s use this capitalist consumption matrix to write the price equation as

p = w 𝓁 + pA + pC^(K)^

Now solve for prices and we get

p = w 𝓁(I - A - C^(K)^)^-1^

We’ve replaced the profit rate with data on the physical distribution of consumption within the capitalist class.

Let {A|~} = A + C^(K)^ and expand the above inverse into the following infinite sum

p = w 𝓁(I - {A|~})^-1^

p = w (𝓁 + 𝓁{A|~} + 𝓁{A|~}^2^ + …)

Now each term of our price equation is a labor input weighted by our augmented input-output matrix {A|~}. As long as we have data on the inter-relationship on flows of capitalist consumptions (just as we need data on the inter-relationship between industry inputs for A) we can use it to calculate our new definition of value.

{v|~} = 𝓁(I - {A|~})^-1^

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Another approach is to set this up as an eigenvalue equation and solve for the eigenvector of non-standard values, but let’s not get into that.

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These augmented terms in our new value equation weigh labor within the infinite sum such that surplus product is now explicitly treated as a cost of production, a separate labor-cost. We now have a measure of value that measures labor-costs in the same manner that prices do. In other words, we have found the costs of labor that prices are measuring!

Prices can be reduced to labor-costs!

p = w {v|~}

The real price of everything … is the toil and trouble of acquiring it.

We just also have to add the surplus labor of producing capitalist’s consumption products to this toil and trouble. For workers to acquire a product, they must not only produce it but they must also produce surplus products somewhere in the economy for the capitalists who employ them. This surplus toil is a necessary institutional cost for capitalism.

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Now one objection I can hear is that real capitalists don’t spend their entire profit on consumption items. ”Sure, this works under the assumptions of simple reproduction, but what about a more realistic capitalist economy?”

I may not be able to completely satisfy your criticisms, but if capitalists are taking part of their profit and investing them in machines, supplies, etc. then these are still products of labor and hence still impose a cost of surplus labor. So capitalist consumption can be expanded to include productive consumption via investments. You could even wrap these investments up in an input-output matrix.

In the case that a capitalist hoards their money, though, then I may not be able to give a satisfactory answer. Perhaps one could introduce a savings parameter that augments their expenditure, and hence the non-standard value.

Here is another interpretation that is something I am working with. It may not be sound, so I am open to debate and better interpretations: We could still see any hoard of money as still representing a claim on surplus value. This is one of money’s functions as a store of value. Whether that value is produced now or in the future, this hoard can still be equated with some mass of surplus labor that would be required for its eventual realization. If the profits are beyond what is feasibly realizable (beyond the economy’s productive capacity) then eventually it will be realized that the profits don’t have value, or at least as much value as previously thought. The claim of surplus value those profits represent collapses and becomes an illusion if that surplus product itself fails to ever materialize.

So excess or hoarded profits and prices may at first appear to be beyond what would our natural prices and our non-standard values predict, but eventually a correction will occur if surplus production can’t be ramped up. In such a case there will be a divergence between the actual market prices and profits vs. what the attractor predicts (and we’ve accepted such divergences from the start) - but the attractor still has a long-term regulation on what is possible and what is actually meaningful. A hoard of gold becomes useless if there’s literally nothing to ever spend it on.

That is, at least, my running theory - so it may not be entirely sound

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Future Directions

Something I like about linear production theory is that it is so open. You could expand this framework in so many different directions depending on your interest.

Nonlinearity?

We’ve been open about the linear assumptions of this model. Perhaps one could keep the network approach presented here, but make it nonlinear by generalizing the production function and how the inputs scale. What are presented as matrices here, like A, would then be nonlinear operators on a network of value flows.

Arghiri Emmanuel and Unequal Exchange

Does the assumption of equal wage rates bother you? Do you want to consider the case where some working sectors are paid less than others? Great, you can do that here!

We can investigate Arghiri Emmanuel’s ideas of unequal exchange between different sectors, or even between a center and a periphery, by keeping wages of the different sectors w~j~ distinct and calculating the flow of value between working-sectors.

Two working-sectors that provide the same labor can still end up consuming different values if the wages in the sectors differ. Higher paying workers end up extracting value produced by lower paid workers, and our linear framework actually allows us to calculate this transfer of value.

This is a different aspect of value transfer in contrast to profits. While profits are value transfers form a working class to a non-laboring exploiting class, unequal exchange is the unequal transfer of value within a class (and/or between nations)

Some will criticize linear production theory for assuming constant wages, but I mean it’s really not hard to introduce a diagonal matrix for distinct wages W into what we’ve done. I don’t understand why economists act like this is some great impossibility.

There are a lot of things I don’t understand about economists.

Global Value Transfers

So we could model unequal exchange within a nation, but we could also extend this entire linear production framework to include different nations each with their own wage rates, labor productivity, etc.

Then you can investigate unequal exchange between the core and the periphery, and value flows between nations and their classes.

This framework also helps to clarify when value is transferred via profits, vs transferred via wage difference. Sometimes Marxists seem to get really confused (and sometimes angry) about unequal exchange. Some think that it claims exploitation is occurring in exchange - but this is not true. Value is still produced by labor in production, as always, but the different prices of labor allow for its unequal distribution.

As Torkil Lauesen states it:

The core of the theory of unequal exchange is the Marxist concept of value. It assumes the existence of a global value of labor on one side, and, on the other side, a historical capitalism, which has polarized the world-system into a center and periphery with a correspondingly high- and low-wage level. This difference in the price of labor entails a value transfer, hidden in the price structure when commodities are exchanged between the center and periphery of the world-system. The central point is not the exchange itself, but the difference between the global value of labor and the different prices of labor power.

Once you can show the above with linear production theory then these confusions start to fade away. The distinction between value transfers via profit and transfers via wage differentials become clearer. You may have to learn some math, but…

There is no royal road to science…

This also gives you a tool to actually calculate it! If you want to do empirical economic research on value flows within imperialism like Jason Hickel (the good one) then here you go!

Reproductive Labor and Healthcare

Here all labor that is done was remunerated with a wage, but reproductive labor is labor and it often doesn’t get paid a wage. I think this model is flexible enough to include this.

Introduce some new worker-nodes specifically for reproductive labor, rewire some relationships between the nodes to reflect the real social relations. Perhaps the reproductive labor nodes don’t get a direct wage but receive value flows from other worker-sectors (typically dominated by male wage-workers with familial institutions) and you have a model capable of giving a voice to unpaid reproductive labor.

You can investigate how much labor they put into reproducing the working class vs how much value they are able to consume given the specific institutional set-ups.

And you can investigate different situations, what if capitalism tries to absorb their labor into commodity producing wage-labor, turning care into a commodity. What if instead the state, or society in some form, were to pay them a wage? One could test out the implications of different social arrangements perhaps.

Here is a materialist framework that allows for the intersection of value theory with feminist economics.

What if we want to model care work more explicitly? Workers should be able to produce a surplus to care for the sick and those who can not work. We can wire up these relations in our network model to model healthcare, hospitals, elderly care, the sky’s the limit. We can even introduce these labor-costs into a new definition of value, one that appropriately accounts for the labor of care.

I think this framework is flexible to introduce all sorts of arrangements as long you are creative and care about asking these questions.

Unproductive vs Productive Labor

We can also introduce different worker-sectors for productive vs unproductive labor as well.

Productive labor produces surplus value, while unproductive labor (like the FIRE sector) redistributes and consumes existing surplus value.

As Paitaridis and Tsoulfidis put it (emphasis mine):

For Marx, productive is the labor, which is activated in the sphere of production, where capital hires labor and non-labor inputs in order to produce more value than the value of inputs… By contrast, in the sphere of distribution, there is no creation of new (use) values, but those that have been already created in the sphere of production change possession or ownership. Similarly, the labor and non-labor inputs employed in the sphere of social maintenance are engaged in the preservation of the existing status quo.

In sum, unproductive is the labor which, rather than expanding production and wealth of society, is simply consuming wealth that has been already produced in order to distribute or protect (maintain) the wealth created in the sphere of production.

We could introduce unproductive labor in financial sectors, or in the police or military, which facilitates profits and maintains the system of exploitation. It maintains the system but doesn’t itself produce value. It maintains and distributes surplus while also consuming a part of it. You could discuss the value flows from productive to unproductive workers in this networked model.

Just make sure to avoid the Jackson Hinkle (the bad one) mistake of abusing this distinction and saying shit like “baristas don’t produce value and are actually exploiting the poor wittle hard-hat hard-working (white) construction workers.”

I have a pit of my own specially dug for those that espouse that sort of shit.

In fact some people prefer to do away with the distinction between productive and unproductive labor because the lines may be hard to draw and can easily be abused by patsocs and other losers. Citing the above paper,

As is well known there is no consensus about the definition of productive and unproductive labor and also about the wealth-reducing effects associated with the rise in unproductive activities and labor.

Are transportation workers producing value? Well on one hand they didn’t directly produce the products, but on the other hand without their labor there is no realization of the product and hence no realization of value. The labor of the transport worker or of the barista is just the final leg of value added in that product's long journey to your pie hole - but value added it still is. So sometimes I see writers classify them as productive and sometimes I see them classified as unproductive.

I am not well versed enough in productive vs unproductive labor myself, so if others want to make a post digging into that then I would benefit and love to read it!

But I do think there are some obvious cases (like police and finance) where the category can be applied and we can discuss the value transfers that such groups suck up. And I think it is useful to be able to distinguish economic growth in productive industries vs unproductive ones like finance. Finance workers redistribute value flows, but also unfortunately financiers have to be fed, housed, clothed, etc.. So they not only redistribute value flows while producing none themselves, but they also parasitically suck some up value.

How about we include as much of that as we can in the above model?

So the flows of value associated with finance are still real flows in the economy - and it’s a flow that can’t directly go toward capital accumulation. So it can be a measured drag on capital accumulation. This the conclusion that Paitaridis and Tsoulfidis make in the above paper:

Unproductive labor is of central importance to capitalism, regardless of the differences (usually minor) in its definitions. The idea is that unproductive activities are a burden to capital accumulation, because they reduce the amount of social product that can be invested productively.

Just be aware of the potential for abuse I suppose?

Toward a Dynamic Model

Another direction is to move beyond discussing the attractor, or gravitational center, and start discussing the actual dynamics.

This is too big of a topic to go into here, so I will leave you with more readings where attempts are made to formalize some dynamic models.

Walker, R. The Dynamics of Value, Price, and Profit (1988) A qualitative description of capitalist dynamics, avoids formal models.

Duménil, G. and Lévy, D. The Dynamics of Competition: A Restoration of the Classical Analysis (1987) An early attempt to model capitalist dynamics. Serves as an inspiration for the following two papers by Wright.

Wright, I. The Emergence of the Law of Value in a Dynamic Simple Commodity Economy (2008)

Wright, I. Classical macrodynamics and the labor theory of value

Now if you want something regarding nonlinear dynamics, you can check out:

Strogatz, S. Nonlinear Dynamics and Chaos

This is an amazing introduction to dynamical systems and an easy read. You will need to know ordinary differential equations, though. Becoming familiar with dynamical systems can also help understand why we took the attractor approach - or why I view the equilibrium system as an attractor. It is recognized that finding analytic solutions for a system’s exact trajectory is often hopeless. You’ll often need to simulate them, and that can be obtuse for the purposes of analysis. Often studying the attractor, the phase portrait, etc. can give you a better intuition for a system than straight up solving the equations of motion on a computer. I think of this framework as operating within this vein.

If you want a (math-intensive) introduction to complexity science then you can try

Thurner, Klimek and Hanel Introduction to the Theory of Complex Systems

It’s a new field though, so there are competing frameworks even within complexity science. The field hasn’t settled down yet. That’s also the case with this field - Marxist economics.

You should also look up other schools such as Temporal Single-System Interpretation; The New Interpretation; authors like M.C. Howard and J.E. King, Dumenil, Levy, Kliman, Moseley, Mohun, Shaikh, etc. Everyone disagrees with each other so it’s a fun time!

Read the economists who interests you, but even more important than that - read Capital!

Some Foundational Texts and Final Comments

There is still much I’ve left out, I didn’t discuss the eigenvalue approach or the research done in that direction such as T. Mariolis et al. Modern Classical Economics and Reality: A Spectral Analysis of the Theory of Value and Distribution.

I didn’t mention anything about joint production, or adding fixed capital (machines) to this. Or adding economic growth. This is covered to various extent in the literature. I’ll leave some texts if one wants to dig in.

There’s always Sraffa’s Production of Commodities by the Means of Commodities (1960) which kick started this. But if you want to dig into some more background for the above then there are some books by Pasinetti I can suggest, but note that as a post-Keynesian he does reject the labor theory of value to a degree. Nonetheless it’s foundational work for the above.

Pasinetti’s Structural Economic Dynamics (1993) is solely focused on the pure labor economy and its growth. Don’t let the fact that it is an abstracted pure labor system dissuade you, though, because there are methods to transform any input-output economy into some equivalent pure labor economy through vertical integration. Once that is done then the results in this book can be applied.

An older paper of Pasinetti’s The Notion of Vertical Integration in Economic Analysis (1973)

An introductory book by Pasinetti I recommend is Lectures on The Theory of Production (1977)

There’s also Essays on the Theory of Joint Production by Pasinetti and others.

And here are some other readings I draw on:

Ian Wright - Nonstandard Labour Values (2008)

Ian Wright - A Category-Mistake in the Classical Labor Theory of Value: Identification and Resolution (2011)

Anders Ekeland - The Transformation problem after Kliman and Wright. Are we approaching a solution? (2012)

Ian Wright - The law of value : a contribution to the classical approach to economic analysis (2016)

Ian Wright - Marx’s transformation problem and Pasinetti’s vertically integrated subsystems (2018)

Ian Wright - The Transformation Problem (2016) Republished in Red Sails in 2023

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That’s about it, I think!

Note that I am pretty limited in what I know, and am still learning, but my goal was to give a thorough introduction to linear production theory so it can be placed in dialogue with other schools out there.

If you don’t agree with it, then at least now you understand it and can better criticize it.

If something about it ticks your fancy, then hopefully I’ve given you some tools to get involved and do your own investigations using it.

Thanks for reading! I apologize for the length, but I hope it was somewhat interesting for whomever made it through to the end.

Now enough interpreting the world, let’s go change it!

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Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

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Price, Value, and Exploitation using Input-Output Tables

Part 5: Prices, Quantities, and Surplus Value for an Example Economy

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Set-Up

Let’s explore the price equation through an example.

Let’s explore an economy with two types of products (two sectors). Product (1) will serve as a consumption item and product (2) will serve as means of production. Product (2) will be used as means of production for its own sector, sector 2, as well as for sector 1. For shorthand, we’ll say product (1) is corn, and product (2) is iron.

The direct labor coefficients will be

.

Meaning that it takes 0.1 hours of labor in sector 1 to produce one “unit” of corn per time-period (perhaps a day, a week, a year, etc.), and 0.3 hours to produce one “unit” of iron in that same time-period. One “hour” here could serve as a stand-in for a hundred hours, a thousand hours, etc., if the small quantities involved bother you.

The input-output matrix will be given by

This means that it takes a~2,1~ = 0.2 units of iron to produce one unit of corn, and a~2,2~ = 0.4 units of iron to produce one unit of iron.

The Leontief inverse is then

The Leontief inverse can also be read directly. The first column tell us that in order to produce a net of 1 unit of corn (product type-1) it takes a gross quantity of 1 unit of corn and ⅓ units of iron. Reading the second column tells us that in order to produce a net of 1 unit of iron requires a gross quantity of 5/3 units of iron.

The standard values of each product are

Reading the columns of v, this means that it would take

• 0.2 hours of labor across both sectors of the economy to produce one net unit of corn (producing corn requires the production of iron),
• and 0.5 hours of labor in the iron sector produce one net unit of iron (Look at A again, iron doesn’t require the production of corn, so all iron value originates from the iron sector)

The value v gives an alternative way, compared to 𝓁 , to slice and dice the labor of the economy. A labor-value v~j~ tells us how much labor across the entire economy is needed for the net production of one unit of good j, i.e. taking into account the labor of its means of production.

Below, we’ll define a consumption vector such that the economy only produces a net product of corn, so only v~1~ is of concern to us here for the purposes of counting labor. If there is no net production of iron we don't need to consider v~2~ when counting total labor.

The Physical Quantities Produced

Let’s keep it simple and say the economy produces one unit of corn for the entire population (it’s a big cob, okay! ). Thus the final consumption for the economy is

and we will keep this total consumption constant throughout our example.

We’ll only change the distribution of this corn between the class. Some portion of corn will be consumed by the working class and the remainder will be consumed by the capitalists so we'll have c = c^(W)^ + c^(K)^.

Because this consumption vector is so simple we can introduce a parameter α ∈ [0, 1] which represents the share of the corn consumed by workers. When α = 1, workers consume all the corn. When α = 0, they consume none.

With this parameter we can define worker consumption as

and capitalist consumption as whatever is left over

And total consumption is the sum of these two,

Note our equation for consumption behaves as we expect. When α = 1 the second term disappears and all of the final product is consumed by the workers while capitalists receive no surplus product. As α approaches zero, though, less is consumed by workers and more is consumed by capitalists. And again, the total final product c is fixed at a constant quantity for simplicity.

We can solve for the gross product required to produce c using q = (I - A)^-1^,

So the total production in the economy is 1 unit of corn and ⅓ units of iron. This produces a net product of 1 unit of corn which is consumed by both classes.

How much labor does this require in total?

Well let’s use L = 𝓁 q, so we have (0.1)(1.0) + (0.3)(⅓) = 0.2 "hours" of total labor.

This required labor will be constant even if the distribution of corn changes between the two classes. Why? Well since c is assumed constant then q, and hence L, will be as well. These quantities (q, L), depend only on the total final consumption c and the technical parameters of the economy (𝓁, A) and not how consumption is divided between the working and capitalist class. So in our example the total labor does not depend on the parameter α since we are keeping c constant.

This also means that value v will not change even as the distribution of consumed corn (α) changes. A portent of the transformation problem!

Solving for Prices

Now let’s look at the price equation and finally solve it

p = w 𝓁 (ϱ I - A)^-1^

where recall ϱ = 1/(1+r).

One approach in solving this, and the one we'll initially follow, is to specify w and solve prices for various profit rates r.

Example 1: Solving Prices for Various Profit Rates

The wage rate w shows up as a simple multiplicative constant and so acts as a scaling factor. We will set it to a constant value of 1 and plot how p behaves for different values of r.

The plot of (log) prices is shown below:

A table of the above prices for select profit rates is also shown below:

When r equals zero, the prices p are equal to w v which are shown with the horizontal red lines in the above figure. You can see that in the table as well. As r increases to its max possible value (in this case r~max~ = 1.5) the prices asymptotically explode toward infinity. Again, this all is assuming that the wage rate is kept constant at w = 1.

This isn’t a dynamic description of an economy. It isn’t stating what the economy will do, how it will move, or how it will behave. It is simply exploring different possible configurations of the equilibrium/attractor state given some parameters.

Since we have calculated prices, we can also show the costs and profits of our system. Let's plot the three following quantities as functions of the profit rate

• Wages: wL = pc^(W)^
• Input Costs: pAq
• Profits: rM = pc^(K)^

We can see that the although the prices do change, the wages paid to workers is constant. We've set w and L to constant values. The wages remain constant, but the amount workers consume must drop off as the prices of the goods rise with the profit rate.

We can also see that the input costs and even the profits explode toward infinity as the profit rate reaches its maximum value at r~max~. The maximum profit rate is dependent on the input-output matrix, and it is the level of profits where the entire net product is consumed by capitalists. Workers receive no consumption items at the maximum profit rate. If workers are still paid a non-zero wage at r~max~ then the only way for workers to receive absolutely zero consumption goods is if prices are infinite, i.e. absolutely unaffordable for workers.

It's best to emphasize that this is asymptotic behavior. Instead of thinking that the maximum profit rate is possible and prices could actually be infinite it is better to think of this as showing the tendency of prices to increase as the profit rates approach r~max~. Again, assuming that wages are constant. A system with maximum profits is itself a breaking point, though, workers would starve and the system wouldn't be able to reproduce itself.

The Relationship between Prices and Physical Quantities

Because the price equation is commonly expressed in terms of A, w and r, it may appear that there is no connection with the physical quantities q and c. But the connections are definitely there and it is surprising to me that some economists working in this framework have failed to make them explicit. Afterall, what are the wages spent on? The worker consumption goods c^(W). Likewise for the profits.

Highlighting this connection also brings us closer to resolving the transformation problem.

Even though the price equation as we wrote it does not appear to explicitly take consumption into account, the distribution of this consumption is implicitly baked in via w and r. I emphasized “as we wrote it” because there is a way to rewrite the price equation in terms c^(W)^ (sort of) and A. I won’t get into it though as it isn't necessary to get into for this post, but there are ways of writing the price equation that makes the connection to the physical system more obvious.

Instead, I would like to go the route of showing how the distribution of the consumption goods, i.e. the surplus product, can be directly linked to the surplus labor that workers must provide. Since the surplus product is bought with profits this provides a direct link between profits and surplus labor.

The following will work nicely because the consumption bundle we’re working with c is relatively simple and easy to inspect. First, recall

pc^(W)^ = wL

Now use our distribution parameter α to rewrite the above as

α pc = wL

α = wL/pc

Note that pc simplifies to p~1~ since c~1~ = 1 and c~2~ = 0 in our example. Solve for α to get

α = wL/p~1~

In our simple example this parameter has a very straightforward interpretation - how much corn the workers consume. One minus this value tells us how much corn the capitalists consume.

Let's now show the relationship between this parameter and the profit rate. To do this we set the wage rate to 1, our same working assumption as before, and solve for prices as a function of the profit rate. We've done that already. So insert those calculated prices for corn into the above equation to get the workers’ corn consumption as a function of the profit rate.

The workers' consumption of corn is shown as the solid red line, while the capitalists' is shown as the lighter dashed line. We can see how the workers’ actual corn consumption falls to zero as the profit rate increases. The profit rate isn't just about money, it's about distribution between the classes. We also see that worker consumption It is equal to 1 (or more generally c~1~) when there is no profit. The value 1-α is the surplus product of corn that capitalists manage to grab.

Now you can relate this above physical graph with the previous two monetary ones. As the profit rate increases and workers are paid the same constant wage wL the prices increase drastically. The workers are able to afford less corn and so their consumption share α begins to plummet. As the profit rate increases the capitalists become the only ones able to afford corn at these absurd prices using their insane profits rM.

We can go one step further now and relate the distribution of necessary and surplus labor to the profit rate. We do this by first solving for v for our example as done earlier

v = [0.2 0.5]

Note that vc is also equal to L, i.e. 0.2 hours. We can use v to split the total labor into that part necessary for the workers’ corn and its required means of production

L~necessary~ = vc^(W)^

L~necessary~ = α vc

L~necessary~ = αL

L~necessary~ = 0.2α hours

and that part for the capitalists’ corn and its required means of production.

L~surplus~ = vc^(K)^

L~surplus~ = (1-α)vc

L~surplus~ = (1-α)L

L~surplus~ = 0.2(1-α) hours

Note that in the above, the total labor L is unchanged. And so are L~1~ and L~2~, i.e. the labor in each sector. Instead, the above is telling us how much of the labor across all sectors is "consumed back" by the workforce vs. consumed by the capitalists. Or, equivalently, how much of the workers' labor across the economy goes into reproducing the workers vs reproducing the capitalists.

When α=1 workers across both sectors work solely for their own consumption goods. The corn-workers are growing corn that only workers will eat, and the iron-workers are smelting iron used in harvesting that corn. Collectively, the workers are able to consume back the labor they put into the economy.

When α=0 none of the labor that workers put into the economy goes back to them via consumption. All the corn the corn-workers grow go to feed capitalists. All the iron the iron-workers smelt go toward machines that harvest corn no worker will ever eat. The workers labor just the same but consume nothing at all when α=0, a biological extreme. Max exploitation.

This happens at the maximum possible profit rate, r~max~=1.5. This is the maximum profit rate because beyond this profit there is simply no more surplus product, and no more surplus labor, that can be squeezed from workers (given that L is held constant). At this point profits have reached a physical limit if they are to be realized as a product to be consumed. Anything in excess would be profits with no goods to buy.

This excess profit would only be money. And you can’t eat money.

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Another experiment we could do is to set the worker consumption of corn to a constant value, and increase the capitalists' consumption via some parameter allowing total labor L to increase as the surplus product grows.

Instead, I’ll leave you with that most dreadful of lines:

This exercise will be left to the reader.

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Prices Revisited: The Numeraire

This approach to solving the price equation we just used where we set the wage rate w and solve for p at various profit rates r is not the common approach in the literature. Instead what is common is to find values of w and r which result in the prices “normalizing” in a specific way. This is done by introducing some vector, or bundle, of physical products b called the numeraire. It is defined such that

pb = 1

or some other constant.

There are different choices for the numeraire, some seem to be more economically useful than others. Some are chosen based on economic reasons, but to be honest, it mostly seems that numeraires which give some nice analytic result are chosen. I honestly don’t like this approach much, but it may be due to my own ignorance and I’m happy to be corrected.

Also, I think that a solution to the transformation problem (which I believe Wright has shown exists) makes the original use for numeraires less motivating - but again, I may be off base here.

There are times where a numeraire of some sort is necessary, though, especially when solving prices via an eigenvalue approach. Since eigenvectors have no length we must normalize an eigenvector representing prices by something unless we're satisfied with the price vector representing only relative prices.

To summarize the idea: if you do set w and r then the prices you get are absolute - the quantities represent an actual price. But an alternative view is to see the price vector as a list of relative prices that we must normalize in some way in order to get absolute prices. This alternative view uses the numeraire to achieve this.

One possible, and economically meaningful, numeraire for this situation is to use b=c^(W)^, and if w and L are known then they can act as constants for the normalization. Your price vector would then be normalized such that the following is satisfied:

pc^(W)^ = wL

But in our previous worked example we already implicitly achieved this normalization when we set w and L beforehand when solving for the prices as a function of r.

Another possible numeraire we could use, though I haven't seen it applied, would be the equation of exchange. We could choose the gross product to act as the numeraire b=q and calculate prices such that the following equation of exchange is satisfied

pq = ℳ𝓋

where ℳ is the total money in supply (not to be confused with M the flow of money-capital) and 𝓋 is the velocity of money. Like I've said, I've never seen this used in practice though, and the velocity of money is hardly ever stable in the first place so it may be hard for ℳ𝓋 to act as a constant for normalizing prices.

Another choice is to use the eigenvector of A as the numeraire. As far as I know this is chosen because it gives a nice linear relationship between w and r. Since prices can’t be reduced to a multiplicative factor of the standard values v, this approach has been used instead. It generates a linear tradeoff between w and r. It is the numeraire that Sraffa used and it is commonly used in literature… but is it economically meaningful? Well… idk.

Just for the purpose of demonstration, though, let’s use the gross product q as a numeraire. I’m not claiming this is a meaningful numeraire to choose, it’s just for demonstration.

For this tactic, we want to assert that the dot product between the prices and the gross product is equal to one, i.e.

Assert pq = 1.

And we want to find the possible values of the wage w and the profit rate r that result in this normalization. Solving for these values of wages and profits gives us the a.) wage-profit curve below on the left and b.) the normalized prices on the right

For all wages w and profit rates r on that line the calculated prices p will result in pq = 1.

We can see that these prices look very different from those we solved earlier, and it's the same equation with the same values of 𝓁 and A! But here we have allowed the wages w to change with profit rates. The prices don't explode toward infinity as profit rates approach their max value r~max~ because wages are no longer held constant and instead approach zero. When wages are non-existent prices no longer need to be infinite for the workers to get nothing.

We can also inspect the costs just as we did before, but now using the normalized prices.

You can see that the cost structure in terms of normalized prices now actually looks more similar to the measures of necessary and surplus labor! So the numeraire method does help in highlighting the distribution of surplus product, but it rests on us trusting that pb is a meaningful quantity.

Setting a numeraires is an alternative way of solving the price equation. It allows us to see how multiple wage and profit rates can result in a similar price structure, but not necessarily the same prices as different combination of values of p~1~ and p~2~ can result in pq = 1 even if q is unchanged. And you can see in the normalized price graph that prices do change even though the normalization pq is a constant.

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Example 2: Using a Specific Wage and Profit Rate

Let’s use one last example to go over the flow of quantities, wages, and profits and relate them to Marx’s quantities used in his circuit. Let’s go over these with one specific combination of the wage rate and profit rate. The specific numbers chosen aren’t meant to be realistic. You can calculate these values yourself as practice.

Let’s take the same example we had above so c is the same as well as A, q, and L.

But now let’s assume the following.

w = $1.00

r = 0.5 (50%)

Using the above values for w and r, and our previous A matrix, prices are

p = [$0.4875 $1.125]

Capitalists initiate a round of production by injecting money-capital M to pay for C, the labor and means of production.

The labor costs are:

wL = $0.2

The costs of the means of production are:

pAq = $0.375

Which results in a total cost of

M = wL + pAq = $0.575

The labor of the workers produces the gross product C’ = q. This is in possession of the capitalists and sold at the above prices.

Capitalists receive M’ = pq from their sales.

M’ = pq = (1+r)M

You can solve the above using either of the two exprssions. Recall that q = [1 ⅓]^T^ if that’s what you use, either way you'll get

M’ = $0.8625

Putting aside the costs M (which will be used in the next circuit of capital) from the revenue M’ leaves a total net profit for the capitalists of

ΔM = M’ - M

ΔM = pq - (wL + pAq)

ΔM = $0.2875

This is can also be found via the profit rate and the total costs, because note that

M’ = (1+r)M = (1+r)( wL + pAq)

And

ΔM = rM = r( wL + pAq)

i.e. capitalist get (1+r)M from sales, they consume by spending rM, and reinvest M back to the cycle to start it anew.

The wages of the workers are used to purchase their consumption goods

wL = $0.2 = pc^(W)^

And the profits of the capitalists are used to purchase their consumption goods

ΔM = r(wL + pAq) = $0.2875 = pc^(K)^

We can find α as before for this set-up to determine the actual quantity of corn in c**^(W) and c**^(K) that each class consumes.

α = wL/pc^(W)^

α = wL/p~1~

α = $0.2/$0.4875

α = 0.4102 (41%)

So workers consume 41% of the final product of corn. Since the total corn consumed is set to a single unit, that means that workers take their $0.2 in wages and purchase 0.4102 units of corn. And equivalently, because of the simplicity of our set-up, we can also say that 41% of the collective labor that workers perform is done for themselves as a class.

Capitalists consume 1-α = 59% of the final product of corn, i.e. they take their $0.2875 in profits and consume 0.5989 units of corn. Again, our set-up implies that capitalists steal 59% of the collective labor of the workers. The remaining money capitalists have is used to restart a new round of production. There is no investment for expanding production in our example so all profits go toward consumption.

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Wrapping Up

Now you've seen an example of how to calculate prices, and you've also seen how the profits can be directly related to the surplus labor workers provide. The price equation first appeared to have this term (profit) that wasn't directly correlated with labor

pq = w 𝓁 q + pAq + Π

but we have shown above that these profits are surplus labor.

A challenge still remains though in finding a measure of labor-value that tracks costs in the same way that prices do. The fact that profits can even be traced back to labor suggests that some type of transformation between labor-costs and price-costs should be possible. But so far it doesn't appear that there is any direct way to get from v to p. If prices p can change as profits increase but values v remain constant, then labor-values can not account for the entire costs which prices represent. But we know, and just saw, that profit is surplus labor!

So each term in our price equation appears to be connected to labor, but there is no way to get p from v alone... What gives?

Maybe there's another way to measure labor? A measure that v just isn't capturing?

Let’s conclude with some final remarks in Part 6.

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Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

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Price, Value, and Exploitation using Input-Output Tables

Part 4: An Economy with Workers, Means of Production, and Capitalists

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Now we are ready to add profit, and hence exploitation, to the mix. Finally, something approaching capitalism.

This will be a model of simple reproduction. There is no economic growth, and the entire net product produced by workers is entirely consumed by capitalists and workers. Nothing is set aside for growth or expanding production; the system reproduces itself at the same level of gross output.

What’s new, in contrast to the previous scheme, is the addition of an exploiting class. Capitalists do not provide labor to the economy, yet they still consume products. Surplus labor must be performed by the workers to produce both these consumption items and the means of production required to make them.

This is important: all of our talk about prices and profits can sometimes mask that underneath it all is surplus labor.

The Physical System

Before diving further into prices, let’s first discuss the physical system. This is nearly identical to the previous model, but now total consumption divides into

• The workers' consumption bundle: c^(W)^
• The capitalists' consumption bundle (a.k.a. the surplus product): c^(K)^

So total consumption is:

c = c^(W)^ + c^(K)^

and the gross product can still be written as

q = (I - A)^-1^ c

as long as we understand that c now contains the consumption for both classes.

Value, Necessary Labor, and Surplus Labor

Our previously mentioned definition of value

v = vA + 𝓁

v = 𝓁(I - A)^-1^

can allow us to invisgate how the workers’ labor divides into the necessary labor (needed to reproduce the working class) and the unnecessary or surplus labor (which goes to the capitalists).

The total labor required for production of all products is L. Value is defined such that the total value of the consumption bundle equals the total labor:

L = 𝓁 q = vc

Now separate c into its class components:

L = vc

L = vc^(W)^ + vc^(K)^

L = Necessary Labor + Surplus Labor

So for workers to produce their consumption bundle and the means of production for their bundle (as well as the means of production for the means of production, and the means of production for the means of production of the means of production, and the …. etc.) they must collectively perform vc^(W)^ hours of labor.

For the workers to produce the capitalists’ consumption bundle and its means of production (and the means of production of its means of production ad nauseum) they must collectively perform vc^(K)^ hours of labor.

Note that v includes embodied labor that 𝓁 alone doesn’t capture, i.e. v contains the labor needed to produce the means of production, the means of production of the means and production, and so on. So we don’t need to explicitly include the labor to produce the means of production (Aq), it’s already wrapped up in the labor value.

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If you want to explore the justification for this definition of value, note:

(I - A)^-1^ = I + A + A^2^ + A^3^ + …

and consider the interpretation of what the n^th^ power of the input-output matrix represents.

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Now for profit and prices.

The Price System

When sector j sells its product, the prices must cover:

• the means of production used to produce product-type j: p a~★,j~ q~j~
• the wages for the labor: w𝓁~j~q~j~
• and also a profit for the capitalists: Π~j~.

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If it helps, recall that a~★,j~ q~j~ is the vector of material inputs required for sector j to product its output at level q~j~.

We can rewrite this as

z^(j)^ = a~★,j~ q~j~

z^(j)^ = [z~1,j~ z~2,j~ … z~j,j~ … z~n,j~]^T^

So the term

p a~★,j~ q~j~

is also

pz^(j)^

which is just the following sum of input costs for sector j:

p~1~ z~1,j~ + p~2~ z~2,j~ + … + p~j~ z~j,j~ + … + p~n~ z~n,j~

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So, following the reasoning of the previous systems we have our initial price equation

p~j~q~j~ = p a~★,j~ q~j~ + w𝓁~j~q~j~ + Π~j~

We’re not done with this equation, though. Let’s rewrite profits as using a sectoral rate of profit r~j~.

The profit rate r~j~ tells us how much capitalists get back in profits relative to what they advance for industry j. Capitalists advance money-capital covering the wages and the costs of the means of production. So the profit rate is defined as the following ratio:

r~j~ = Profits / Money-Capital Advanced

r~j~ = Π~j~ / ( p a~★,j~ q~j~ + w𝓁~j~q~j~).

Rearrange this to get an expression for profits

Π~j~ = r~j~ ( p a~★,j~ q~j~ + w𝓁~j~q~j~).

Sraffa, and some others, exclude wages from the advanced capital, but we’ll stick with the above as it’s closer to Marx and the classical economists.

Substitute this definition of profit back into the our initial price price equation above

p~j~q~j~ = (1 + r~j~)(p a~★,j~ q~j~ + w𝓁~j~q~j~)

and divide by q~j~ to get:

p~j~ = (1 + r~j~)(p a~★,j~ + w𝓁~j~) .

Now, assume capital is feely mobile and competition equalizes the rate of profit across all sectors. If one sector has a lower profit rate than the average then capitalists disinvest, labor exits, supply shrinks, and prices rise. This causes the profit rate to increase and restimulates investment in that sector. This is one institutional mechanism underlying the law of value.

The a.) movement of capital from sectors with low to high rate of profit, and b.) the effects of supply and demand on actual market prices are mechanisms that equalize profits and also reallocate labor.

We aren't modeling these market price fluctuations due to supply and demand, though. We’re interested in attractors, not actual trajectories. Again, think long-term prices, or regulating prices. So note that this profit rate r is the regulating center of the observed rates — not the actual momentary measurement existing at any given time due to existing market prices.

Using a single profit rate across all sectors is analogous to how we used a single wage for all workers. If you want to resist this assumption and keep each distinct sectoral profit rate r~j~, then you’ll need to introduce a diagonal matrix of profit rates R.

When, or if, there is one regulating profit rate for the economy then we can drop the index for the profit rate,

r~j~ = r,

Now, just as we’ve done before, let’s turn our price equation for one sector into a vector price equation for all sectors’ prices:

p = (1+r)(p A + w 𝓁).

This is our equation for natural prices in a system with exploitation but no growth.

Let’s solve it now for p to get

p = w 𝓁 (ϱ I - A)^-1^

where ϱ = 1/(1+r).

Note that this is close, but not identical, to the the price equation discussed last time when there was no exploitation: p = w 𝓁 (I - A)^-1^ = w v.

The Transformation Problem

As r → 0, the system with exploitation collapses to the non-exploitative case. But as r increases, p shifts while v still remains unchanged. This is the core of the transformation problem when expressed with input-output analysis. Surplus distribution affects prices but not values (as conventionally defined).

Just to say this again to be clear: as the profit rate r changes our definition of value

v = 𝓁 (I - A)^-1^

is not impacted. Each of the above quantities in the value equation are set in our example and they don’t change as more or less product goes to the capitalists. The profit rate doesn’t impact 𝓁 or A, and hence v is invariant as r changes.

But the prices

p = w 𝓁 (ϱ I - A)^-1^

do change as the profit rate (or ϱ) changes.

If v is invariant to changes in the profit rates but p isn’t, then we have a problem if we want to claim that prices can be expressed in terms of value.

Don’t panic, though. There is a way to address this problem head on without hiding from it or throwing out the use of input-output tables. Unfortunately I won’t do it justice in these posts, but in the next post I will still show the relationship between prices, profit rates and surplus labor using an example. And the fact that such a relationship can still be shown to exist hints that labor is still there, somewhere, in the prices.

Before solving the above price equation with an example, though, let’s discuss the circular flow of the economy in this exploitative case.

The Circular Flow and a Discussion of Profit

(Note: this is a reformulation of Marx’s M → C (L + MP) … P… → M’)

As before,

• Workers contribute L hours of labor

• In return, they receive wages wL.

Capitalists inject money-capital to pay for

• Wages: wL

• Inputs: pAq

This injection of money capital can be written as M using Marx’s notation, and we can also connect it to the input-output quantities via

M = mq

M = pAq + w 𝓁 q

where m = pA + w 𝓁 are the unit costs.

Don't confuse M with the total money supply. It is not a stock of total money in the economy, instead it's the flow of money-capital.

Workers make the product q, but capitalists own it and sell it in three parts:

• z = Aq back to industries as means of production

• c^(W)^ to workers as means of consumption

• c^(K)^ back to capitalists as their means of consumption.

Under simple reproduction there is no growth, so no investments to increase capacity. The same level of means of production z are invested during each round of production.

When capitalists sell the total product at natural prices, they receive a flow of revenue equaling:

M’ = pq

From this, they set aside a part to advance another round of capital. In simple reproduction, this new round of capital is identical to the previous one: M = m q = pAq + w 𝓁 q.

After advancing capital M and receiving M’ = pq from sales, they still have (they hope) a net positive flow of money coming into their pockets which is equal to

M’ - M = pq - pAq - wL

This positive net in-flow is the total profit Π of the capitalists, and using the definition of the profit rate we can write it as

Π = rM

Π = r mq

Π = r(pAq + wL)

This profit Π is also written as ΔM using the more classical Marxist notation.

Capitalists aim to acquire c^(K)^ without laboring. Hence, the system must compel workers to engage in surplus labor for the production of this surplus product that capitalists consume.

The profits that capitalists make are spent on purchasing this surplus product.

Let’s discuss what each class does with the money distributed to them.

First, let’s discuss the industries - not really a class, just a section of our model. In the aggregate, industries (or the capitalists that own industries) sell pAq to other industries. This quantity of money flows out of industry but then right back into it. It’s like a closed loop.

Now the workers. Capitalists hire workers for their labor-power. The workers can’t check out whenever they want - capitalists make them work for some specific time. The length of the working day isn't a natural law, though, it's part of the class struggle. Organized labor can fight for a reduction in the working hours, and fight for increases in pay as well. This isn’t mechanical, it’s determined by our struggle.

Now, as long as there are capitalists with power and self-preservation, the wages they pay to workers are only enough for them (in the aggregate) to purchase their means of consumption

pc^(W)^ = wL.

The workers can not use their wage to afford the entire consumption bundle c that they've created: c^(W)^ < c. Workers (collectively) are forced to produce the entire final consumption bundle when employed, but are never able to buy it back. Workers must labor for more than they could ever equivalently receive back in value via consumption.

I.e., the workers must give L to produce a final product of c

vc = L

But the value of what they receive through consumption is less than the labor they originally worked!

vc^(W)^ < L

And so a surplus of the final consumption still remains:

surplus product = q - Aq - c^(W)^

As mentioned, the surplus is produced by workers but can’t be afforded by them. And it isn’t needed by industries as means of production. Where does it go?

This surplus product

c^(K)^ = q - Aq - c^(W)^

is purchased as means of consumption by capitalists with their profits. So

pc^(K)^ = Π

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At at the risk of sounding pedantic, let’s just summarize the various ways we can write out profit

Π = ΔM = rM = r mq = r( wL + pAq)

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In our simplified case of simple reproduction all of the capitalists’ profit goes toward purchasing their consumption items.

Note that the profit rate r we are using is also Marx’s profit rate, but expressed with monetary quantities and not as values (as standardly defined). We can’t yet equate prices with value so r is in terms of money. Again, the transformation problem between values and prices (and its potential solutions) really deserves its own post.

Similar to how Marx wrote the profit rate as surplus value over constant and variable capital

r = S/(C + V),

we are writing it as

r = pc^(K)^ / (pAq + wL)

It’s slightly different, but still note

• pc^(K)^ is the profit Π which is the monetary expression of the surplus product

• pAq is the monetary cost of means of production, or constant capital

• wL is the monetary cost of labor-power, or the variable capital

The above has just been another way of expressing Marx’s circuit

M → C (L, MP) → …P… → C’ → M’ = M+ΔM

Summary of the Circuit of Capital

Here’s that connection summarized one last time:

• Capitalists advance money-capital:

• M = wL + pAq

• And use it to purchase C which consists of labor L and inputs MP = Aq

• Laborers transform this into output

• C’ = q = Aq + c^(W)^ + c^(K)^

• And capitalist sell this output to receive

• M’ = pq = M + ΔM = M + rM

• They then reinvest another round of M and take home ΔM = r( wL + pAq) to purchase their consumption items, i.e.

• ΔM = pc^(K)^.

We have our familiar circuit of capital

M → C (L, MP) → …P… → C’ → M’ = M+ΔM

This process rests on capitalists owning the means of production and hence the production process. Capitalists can compel workers to provide surplus labor beyond what is needed for their own reproduction.

Summary for Value and Surplus Labor

Capitalists force workers to provide L total hours of labor.

L = vc

L = L~necessary~ + L~surplus~

Part of this total labor is L~necessary~ which is used to produce the workers’ own consumption items c^(W)^

L~necessary~ = vc^(W)^

And another part of the total labor is used to produce the capitalist’ consumption items c^(K)^

L~surplus~ = vc^(K)^

Workers are paid only enough to reproduce themselves with c^(W)^, and capitalists appropriate the surplus product c^(K)^.

A Final Note on Natural Prices

Now that we have a system of class exploitation, this class will want to preserve itself. Natural prices are now those that allow the capitalist class, and the capitalist system as a whole, to reproduce itself. Earlier, without the capitalist class, the natural prices were such that allowed for reproduction of only workers and the means of production. But here the natural prices of capitalism are those that allow for capitalists to sustain themselves at some level of profit.

This profit, and hence the distribution of labor into necessary and surplus, is determined by the class struggle. It isn’t mechanistically determined. Our model allows us to see how prices and profits are related, but by itself it doesn’t tell us what these profits are set to.

Also, it isn’t as if capitalists know what these natural prices are and set market prices to them. Instead an actual institutional mechanism must exist to “discover” these natural prices. Under capitalism it is a market mechanism of some sort that drives market prices toward natural prices.

In early capitalism there was much more competition between firms, and competing capitalists setting prices in the market would lead to the emergence of market prices settling around some natural price. Again, it is a somewhat objective process, the emergence of natural prices is beyond the will of any one individual capitalist. When monopolies form, then there is more control that one giant firm can have on market prices - and hence on the natural prices. But even then, monopolies do not mean that there is no competition (until the world is totally dominated by the Weyland-Yutani I suppose?)

If a capitalist sells their goods at market prices way below the natural price, then that results in less profit for them (and any shareholders) unless they also severely undercut their workers. That creates a pressure for them to raise their prices closer to natural prices unless they can sustain the lower prices through technological improvements.

Now, if they do make a technological breakthrough then these firms can sustain their lower prices and may tend to dominate the market. But their technological improvements and new dominating presence lowers the labor coefficients, lowers the socially necessary labor time, and has a downward pressure on natural prices.

In the opposite case, if a capitalist sells their goods at market prices above the tending natural price, then sure they make more profit, but they aren't competitive. So fewer sales. Smaller marker share. Bankruptcy is likely to follow unless they can lower prices to stay competitive.

The inner mechanisms and dynamics of market prices, how they converge to natural prices, and how the law of value also leads to technological improvement and the tendency of profit rate to fall isn’t discussed by this model. But maybe you can see ways of including them. But I also don't think it's this framework's intention to erase this dynamic side of the economy. In the last post I'll briefly discuss some work that has tried to breath movement this framework.

Also, there is a range of natural prices for our system depending on how much surplus labor the capitalists are able to extract from us. And you can see that with the equation, the natural prices vary with profit rates even when the technical coefficients (𝓁, A) don't change.

We’ll explore this in more detail when we solve the price equation with an example. On to Part 5!

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Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

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Price, Value, and Exploitation using Input-Output Tables

Part 3: A Worker-Only Economy with Means of Production

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Let’s add in the input-output matrix.

Now we can start to include means of production into the system as circulating capital. We will not yet add profit and so no exploiting class. This system will allow us to introduce the input-output matrix.

The Input-Output Matrix

The input-output matrix is a tool which quantifies the material connections between different economic sectors. It tells us how many products each sector will require from all others when producing a good.

The set-up is almost identical to before, except that in order for each industrial sector j to produce its gross output q~j~ it must now use products produced by other sectors. The products that sector j requires as means of production can be expressed in the n x 1 column-vector z^(j)^, or sometimes written as z~★,j~. The star ★ in the row-index signifies that all sectors are included as n rows in the column-vector z~★,j~.

To clarify, sector 1 requires some quantity z~1,1~ of product-type 1 (produced by sector 1), some quantity z~2,1~ of product-type 2 (produced by sector 2), and so on. These are all encoded in sector 1’s necessary means of production vector z^(1)^.

Sector 2 similarly requires z~1,2~, z~2,2~, etc., forming z^(2)^.

More generally, sector j requires:

z^(j)^ = [z~1,j~ z~2,j~ … z~j,j~ … z~n,j~]^T^

(Note the transpose).

We continue to assume linearity (constant returns to scale), so we divide each quantity z~i,j~ by sector j’s output q~j~, giving us the inter-sectoral input coefficients a~i,j~

a~i,j~ = z~i,j~/q~j~

or equivalently

z~i,j~ = a~i,j~ q~j~ .

(Helpful Note Pay attention to index order in two-index quantities. The order helps track the flow of inputs. For example z~i,j~ is the quantity of commodity i that flows into sector j. These are of product-type i, so they must be priced at p~i~. We'll see that the money that sector j must pay to sector i is then p~i~ a~i,j~ q~j~.)

The quantities a~i,j~ can be placed in an input-output matrix

which can also be written with the single generic term using index-notation

A = [a~i,j~].

A matrix can be thought of as a table of numbers where i indexes the row, and j the columns. For example, the third row and fourth column of A is denoted as a~3,4~. In our case there are at most n rows and n columns in our matrix. A matrix isn't just any table, though, has particular rules for how it multiples, adds, etc. with other tables. You can add two matrices if they're the same size (same number of rows and column). You can even take a matrix and multiply it by a vector as long as certain rules are met. We won't get into all the rules of linear algebra, there are plenty of introductions to that field. For the rest of the post I'll aim to give enough context to guide you through the math without explicitly explaining it all.

Economically, a~i,j~ represents the quantity of product i directly required in the creation of one unit of commodity j. This matrix must be non-negative (it can't take negative quantities to produce another), and for the economy to be productive A must also satisfy the Hawkins-Simon Condition.

We can also interpret A as a network of flows between sectors.

(n=3 industry sectors and the inter-industry coefficients between them. For example, in order for industry 3 to produce a single output it requires a~1,3~ from industry 1 and a~3,3~ of its own product to serve back as inputs.)

The Physical System

Let’s now describe the physical equations. As before, we treat the final consumption vector c as given and solve for the gross product q.

In equilibrium, sector j must produce enough to cover the desired final production c~j~, plus enough to serve as means of production for the entire economy. The total quantity it must supply to other sectors to serve as their inputs is z~j~.

So the gross product of sector j is

q~j~ = z~j~ + c~j~

We can relate z~j~ to the means of production requirements of the other sectors using the input-output matrix. Let’s express z~j~ in terms of other sectors’ needs: sector 1 requires z~j,1~ from sector j, sector 2 requires z~j,2~ from sector j, and etc. So:

z~j~ = z~j,1~ + z~j,2~ + … + z~j,n~

Substituting the input coefficients give us

z~j~ = a~j,1~ q~1~ + a~j,2~ q~2~ + … + a~j,j~ q~j~ + … + a~j,n~ q~n~

Now here’s the magic: note that the above is the j^th^ row of input-output matrix A times the column-vector q.

So the gross product for sector j can also be written as

q~j~ = a~j,★~ q + c~j~

where a~j,★~ is the j^th^ row of the input-output matrix A. Here the star ★ in the column-index means that all sectors are included as n columns in this row-vector.

Note that the equilibrium output of sector j depends on the produced outputs of all other sectors (given in q and related via the input-output matrix). So A helps to encode the inter-relationships between the sectors.

Since the above is true for any sector (again j is arbitrary) we can write our vector equation for the gross output of all sectors as

q = Aq + c

Since we take c as given data, we can solve for the gross output q

q = (I - A)^-1^ c

The matrix (I - A)^-1^ is commonly called the Leontief inverse, and it must exist for the matrix A to be considered productive.

The Price System

To find the price equation we can think about the costs that a sector j has when it produces its output q~j~.

Labor Costs: First it has the labor costs which we’ve previously discussed, w 𝓁~j~ q~j~.

Means of Production Costs: But now it must also be able to afford the means of production from other sectors. As discussed earlier, for sector j to make q~j~ it must productively consume a vector of products given by the n x 1 column-vector z^(j)^ = [z~1,j~ z~2,j~ … z~n,j~]^T^.

For this, sector j must pay an amount p~1~ z~1,j~ to sector 1, an amount p~2~ z~2,j~ to sector 2, …, etc.. Recall that z~i,j~ is a good of product-type i produced by sector i that flows to sector j. Since it is of product i it must be sold at sector i prices.

So the costs that sector j incurs to afford its means of production are

p~1~ z~1,j~ + p~2~ z~2,j~ + … + p~j~ z~j,j~ + … + p~n~ z~n,j~

(An example of inputs and costs for some random sector 2)

Exploiting linearity, we can introduce the inter-sector coefficients again and write the above as

p~1~ a~1,j~ q~j~ + p~2~ a~2,j~ q~j~ + … + p~j~ a~j,j~ q~j~ + … + p~n~ a~n,j~ q~j~

or

(p~1~ a~1,j~ + p~2~ a~2,j~ + … + p~j~ a~j,j~ + … + p~n~ a~n,j~) q~j~

Note that the terms in the parentheses are a multiplication of the row vector of prices p times the j^th^ column of the input-output matrix A. So we can write the above as

p a~★,j~ q~j~

where a~★,j~ is the j^th^ column of input-output matrix A.

The price of product j when it sells q~j~ must cover these costs. So,

p~j~ q~j~ = p a~★,j~ q~j~ + w 𝓁~j~ q~j~

Divide both sides by q~j~ and we have our price equation for sector j

p~j~ = p a~★,j~ + w𝓁~j~

(An example of the unit-inputs and -costs of a three sector economy.)

Finally, note that the above applies for any sector j, and hence any column in A. So we can write the prices for all sectors as the following vector equation

p = pA + w 𝓁

And we can solve this for p to get

p = w 𝓁 (I - A)^-1^

Again, we rarely discussed the institutions involved in distribution and exchange. We can think that p represents natural prices that real market prices tend to gravitate toward for the system to have long-term viability. But perhaps there is no market mechanism and we have a type of planned economy that still keeps track of costs in this manner. A planned mechanism that keeps track of costs as labor would be nice as well, because then we could determine how much labor to reallocate across an economy to meet demand.

Value and Prices and the Fundamentals of Labor

Earlier we said that direct labor coefficients in 𝓁 only capture the immediate labor for each product. They do not account for the labor embodied in the inputs.

We can define value as the embodied labor of a product, i.e. a way of measuring the indirect labor required throughout the entire economy for the production of a commodity.

Pasinetti (1988) and Wright (2019) show there are multiple possible value definitions depending on how we define what is meant by “embodied”. For example,

• Do we want to count the labor required just to reproduce the good and its means of production?
• Do we want to count the above and and any extra labor required to grow the economy at a steady rate?
• Do we want to count both of the above and the labor required to produce the capitalists’ consumption goods?

The language of "embodied" has some ambiguity when we try to write it down as an equation. But let’s stick with the standard and common definition of value - the labor directly involved in production and the labor embedded in the means of production. To you give a brief taste, without deriving it, the standard measure of value is

v = 𝓁 (I - A)^-1^

where v is a row-vector of values for each product j.

Vector v contains the values of each unit product, i.e. v~j~ is the value of a one unit of product j. The total value of x~j~ units of product j is then v~j~ x~j~.

Notice that in our system of workers and means of production (with no capitalists), prices are proportional to these standard values.

p = w 𝓁 (I - A)^-1^

p = w v

The true cost, the real price … is still labor. But now we don’t measure labor using the direct labor coefficient, we must now move to using a measure of labor that takes into account the costs of the means of production, i.e. value as we’ve defined it.

But, as we will see, once exploitation is introduced prices do change as profits increase or decrease although values as defined above are not impacted. A transformation problem arises between these values and prices.

Some, like Samuelson, have argued that since values aren’t necessary to determine prices, they’re unnecessary altogether (”get that pesky Marx out-a-here!,” they say). Others, such as Wright (2019), building on Pasinetti (1988), have constructed definitions of value that are commensurate with prices. But for the most part this transformation problem has been a thorn of some sort, some people ignore it, some try to solve it, some say that it’s a waste of time. Some think it’s the Marxist version of debating pin dancing angels.

But I do think that talk of the transformation is important, and as this current demonstration continues it can given the impression (even to some “Marxian” economists) that value is redundant, unnecessary, and can be discarded.

The neo-Ricardians, post- and neo-Keynesians, and most "intelligent" economists at university certainly deem it as unnecessary baggage from that crazy ol’ Marx. Some Marxists even abandon hope and think labor values are irrelevant and give up on the labor theory of value. Some become a strange type of anti-empiricist who turn away from a scientific (and quantitative) approach to political economy. I think these are serious errors.

The following isn't necessarily part of the standard framework, but I want to rant and soapbox here and emphasize, without yet going into the math of it, that value is important as it is what grounds economic measures. It ties measures back to labor, the root of the economy. If there is no labor, and no labor allocation via the law of value, then there are no reproducible commodities. And hence there is no long-term economic system. We'd consume only the use-values that nature provides. Humans are not the type of organism who does this. You can see Shaikh (1982) for more on this.

Labor is foundational to economics, despite the attempts by economists to hide this. But this foundation is even deeper. Labor not only made the world, but as Engels argues labor made us human, i.e. even our biological evolution as a social species was done through labor. It is fundamental to our social-being, or our species-being.

If this isn't a glowing affirmation of the working class and its potential, then I don't know what is.

Any future existence of our species will hinge on how our social-being makes use of our laboring capacities. Climate change provides a clear example. Will we rise to the challenge and rearrange our society and production in order to prevent ecological collapse? Or will we make endless Funko pops till we die?

The Circular Flow

Before you panic, let’s recap the flow of labor, products, and money for this modeled economy.

• Workers provide total labor L, which is the sum of L~j~ from each sector j.

• Some of this labor goes into producing their consumption items c, some to producing means of production z = Aq.

• Regardless of the type of work, workers (in the aggregate) receive a wage wL for it.

• Workers use their wage to purchase their consumption items c, so pc = wL

• Industries use part of the product to produce inputs z = Aq, and these input costs sum to pAq. In the aggregate, these costs go out toward the industries and are received back by them.

Another Take on Natural Prices

Here is another structural way to think of natural prices. Note that if the natural prices were lower than what the system dictates, it may be possible for workers to purchase more consumption items, i.e. they could consume more than c. If this occurs then their consumption would be “eating into” the products that must serve as means of production. This would reduce the remaining product available as z, making the system unsustainable in the long-run unless workers can supply more labor to produce more means of production.

In other words, assuming all else was left constant (wL doesn’t change, and L and q doesn’t change) if p were lower, then c could increase but that would cut into q and cause a reduction in z which would be unsustainable for the production of c in the first place! (Remember, q = z + c).

So, the natural prices are such that allow for workers to eject c < q from the flow of commodities and ensures that z remains for future reproduction. The system can reproduce itself!

Another way of expressing it: the natural prices are those that take into account the labor costs required to produce the means of production, and so they result in prices which ensure that those supplies are set aside and not consumed with the workers’ wages.

These natural prices are not about markets per se, they represent reproduction-consistent values that close the loop on production and consumption.

Unfortunately, in the next post I won’t get into the solution for the transformation problem of how prices and values do relate (keep waiting). But I will clarify it. If interested, you can check out Wright (2019) for how this framework can resolve the transformation problem. At the very end at post 6 I'll give a taste of the solution.

Before that, I'll use this framework and it’s definition of value to show directly how profits are exploited surplus labor.

But first, let's talk introduce exploitation and discuss its impact on prices in Part 4

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Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

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Price, Value, and Exploitation using Input-Output Tables

Part 2: A Pure-Labor Economy

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Now let’s begin with the simplest toy model: a pure labor economy.

The Set-Up

Suppose we have n economic, or industrial sectors, each producing one type of product indexed by j. Each economic sector has an associated worker-sector, which embodies all the N~j~ workers in that industry. We will not consider joint production, which is where one sector can produce more than one type of product.

(The set up with n industry-sectors and n worker-sectors)

If there are N~j~ workers in sector j, each working h hours per time-period, then the total labor-hours per time-period of sector j are N~j~ h, which we define as L~j~. We can think of each worker-sector as providing L~j~ hours to its industrial sector j per time-period. The units of h and L~j~ are flows: hours per unit of time. That may sound odd, but it’s familiar. We often talk about labor as “8 hours per day”, or “40 hours per week”. As long as the units are consistent across quantities, the time-period in question can be measured in any way. Typically it's determined by the time resolution of the data. In input-output analysis, I often see labor measured in hours per year.

In this framework, we distinguish two dual systems:

• The physical system, describing the flow of real products
• The price system, which we’ll cover next.

The Physical System

Let’s start with the physical system. In a pure labor economy, it’s simple: each j^th^ worker-sector contributes L~j~ labor hours to its sector, which produces a flow of q~j~ goods (of type j) per time-period. The units of q~j~ are in “units of product-j per time-period” (e.g. tonnes of iron per year, or pounds of apples per month, etc.).

If we denote the mean-productivity of sector j as ξ~j~ (units of product j per hour of labor) then:

q~j~ = ξ~j~ L~j~.

This assumes a linear relationship between labor input and output. More commonly, we use the direct labor-coefficient 𝓁~j~ which indicates how many labor-hours are needed to produce one unit of product j. For example, if it takes workers in a car industry 4 hours to build a single car, with the parts on hand, then 𝓁~car~ = 4 hours. The units of 𝓁~j~ are labor-hours per unit of product, and it’s the reciprocal of productivity:

𝓁~j~ = 1/ξ~j~.

We can gather the labor coefficients of all sectors in a 1 x n row-vector:

𝓁 = [𝓁~1~ 𝓁~2~ … 𝓁~n~]

Using our technical labor coefficient and rearranging our equation between gross output and labor time gives us the following important relationship

L~j~ = 𝓁~j~ q~j~

This tells us the labor in sector j given its output.

Note that 𝓁~j~ does not include the total embodied labor from other sectors that are needed to produce product j. For our car example, 𝓁~car~ excludes the labor needed to produce rubber, glass, steel, etc.. It only includes assembly occurring in the auto sector. Embodied labor is instead measured by value, and there can be different measures of value depending on what you mean by “embodied labor”. We’ll touch on value later, though. For a pure-labor economy, 𝓁 suffices as there are no intermediate inputs, an idealized abstraction for sure.

(Each sector j provides L~j~ hours of labor to produce a gross output of q~j~)

Often we don’t need to split the working class into n worker-sectors, and we can instead treat workers collectively as one big (happy) worker-sector. It depends on what level of resolution we're interested in for the question at hand.

Now if the labor in sector j is L~j~ = 𝓁~j~ q~j~, then we can sum the labor across all sectors to get the total labor in the economy:

L = 𝓁 q

This is a dot product of two vectors, 𝓁 and q. This is just a convenient way of expressing the sum of 𝓁~j~ q~j~ for all n sectors.

What is q? Well it's a n x 1 column-vector of produced goods for each sector:

q = [q~1~ q~2~ … q~n~]^T^

In this framework, we often treat the net product (here simplified as just final consumption) as given, and solve for the gross product q. In a pure labor economy, this is trivial—all products are directly consumed by workers:

q = c

Where c is an n x 1 column-vector of total consumption

c= [c~1~ c~2~ … c~n~]^T^

Each element c~j~ is the total amount of product-type j that is consumed by all workers regardless of what sector they may be working in.

(The system of production and consumption.)

Here because we are not interested in the exchange and distribution of products between workers, we can treat all workers as a single node and assume that this exchange succeeds under some ideal conditions.

If, or when, we are interested in this distribution between workers then we could subdivide consumption further. For instance, c~i,j~ could represent how much of product i is consumed by workers in sector j and we could use these to create a consumption network, but we’ll save that for another time.

The Price System

Now we can move into the price system for the pure-labor economy.

Suppose workers in sector j are paid a wage w~j~ per hour. For simplicity, we assume a uniform wage w across sectors. This could represent an average wage, or result from labor mobility equalizing pay across sectors (given suitable institutions). If we drop this assumption, we’d have to use a diagonal matrix W encoding wages w~j~, and supply data or theory for their variation.

The total wage paid out to the workers of sector j is wL~j~, and the total wage paid to all workers is wL.

A Note on Institutions In this pure-labor economy, there's no exploiting class. We might imagine this economy as:

• A system of independent, or petty, producers paying themselves a "wage" from their revenue.
• A system where labor and property is completely socialized. All workers get paid a wage for their performed labor from the institutions (the state, co-ops, etc.)

The specific institutions are left vague, but if some institution is at play then it's treated as ideal, i.e. it is working smoothly and is self-correcting enough that we can ignore its inner-mechanisms. Real, messy, institutions must obviously exist to do the actual organizing of people and the economy, but a "pre-institutional" lens is our attempt at trying to abstract away form the details of the intuitions and to try to find something more general.

Another thing about institutions, a pre-institutional lens can't be fully abstracted away from specific institutions. Aside from the institutions that organize the productive units, we still posit some system of wages and monetary flows. Also we aren't discussing the market institutions, or even assuming that a market is what exists to give these products prices. But something is still acting as a means of accounting for costs, at least (here the true costs are labor). Perhaps there is an ideal market that is finding our natural prices, or perhaps there's a system of socialized distribution that keeps track of prices as purely an accounting mechanism. Again, we're trying to be as "pre-institutional" as we can (if that is possible). This framework doesn't really touch on the details of exchange, other than assuming that it proceeds, it isn't the focus.

Now before you walk away in disgust, though, I'd like to suggest that this approach leads to something like an immanent critique. If there's a market, then sure we'll take on the assumptions of a "perfect market" where supply can meet demand and the market price corresponds to the price of production/natural price. If there's a labor market, then sure we'll take on the assumptions that labor can smoothly move across sectors. We'll meet these assumptions and still show that the system is one of exploitation. We'll still show that profit is stolen labor even when the market is behaving as it should - even when the institutions are perfect.

Crony capitalism isn't the problem, it's just capitalism.

But not here in Part 2 just yet, let's stick with pure-labor before getting carried away.

Continuing, since there is no profit and also no capital inputs, each sector’s revenue, p~j~ q~j~, must just cover its wage bill:

p~j~ q~j~ = w L~j~

p~j~ q~j~ = w 𝓁~j~ q~j~

Dividing both sides by q~j~,

p~j~ = w 𝓁~j~

This holds for all j, so in vector form we have:

p = w 𝓁

where p is a 1 x n row-vector for each sector’s natural price.

Note that the physical system is given by column-vectors (q, c) and the price system is given by row-vectors (p). This convention, while not universal in the literature, helps express their duality and keeps the math clean.

Note that here in our pure labor economy the real cost of an item, its price, is the labor required for producing it, scaled by some wage-rate. Here, Adam Smith's quote rings out as obvious:

The real price of everything, what everything really costs to the man who wants to acquire it, is the toil and trouble of acquiring it.

We'll see though, that introducing profit muddles this a bit and results in something called the transformation problem. It doesn't mean that labor isn't involved in profit (it is and we'll show that), but it does mean that the direct link between labor and prices appears to break down. The real price of everything appears to not just be the toil and trouble of acquiring it, but instead that and a little bit extra.

When the transformation problem was pointed out our vulgar economists couldn't pass the chance to swoop in like vultures and interject that it was the toil and trouble of the poor bourgeoise that was that little bit extra. Fair and square.

But noticed I said appears. We can show, using this exact framework, how the cost of labor comes back into play even with profits. The connection between values and prices, i.e. costs, can be restored. Labor is the natural cost of something and origin of its value, even when capitalists make profit. But restoring that direct connection requires a little openness in thinking of how we define value for the question at hand. I won't get it into it, even in part 6. But I will most all of the pieces there for you. And I will show how you can still relate the profit rate directly to the amount of surplus labor extracted. That little extra on prices and values, perhaps in the future I'll come back to it.

The Circular Flow in a Pure Labor Economy

For now, Let’s summarize the flow of labor, products, and money in a pure labor economy.

• Workers supply labor L~j~ to sector j, summing to total labor L

• They receive total wages wL.

• They spend their entire wage on consumption c, which collectively costs them pc

Assuming no savings or debt, we have:

pc = wL

(The monetary flows involved. Wages are distributed to the working class for their labor and they spend it back on consumption items)

For purposes of visualization, we can further aggregate all n sectors together and represent this circular flow between labor, quantities, and money.

For the flow of labor and products we have

And for the flow of wages and payments we have

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In Part 3 we’ll add circulating capital to this and introduce the input-output matrix.

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Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

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Price, Value, and Exploitation using Input-Output Tables

Part 1: Introduction

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This is a long post, but it is bouncing off a discussion. prompted by iie [they/them, he/him]. I also want to give a shoutout to CyborgMarx [any, any] and the other posters in the original thread as they had great contributions. The predecessor thread is What do Marxist economists say about the role of supply and demand?.

This post will go through a detailed overview of the equilibrium state of an economy using input-output analysis with some insights from Marx.

It will start with a pure labor economy where the only inputs are labor, before developing it into a description of simple reproduction with a capitalist and working class.

The main parts are

• Part 1: The Introduction
• Part 2: A Pure Labor Economy
• Part 3: An Economy with Circulating Capital, but no Exploitation
• Part 4: An Economy with Circulating Capital and Exploitation (Simple Reproduction)
• Part 5: Two Examples where we solve for prices, quantities, etc.
• Part 6: Conclusion and some Future Directions

It requires some familiarity with linear algebra, but I'll step through the math to make it as accessible as possible. This framework stems primarily from Piero Sraffa’s work, Production of Commodities by Means of Commodities, with contributions by Luigi Pasinetti and Ian Wright.

I’ve also included an attempt to bring Marx back into the picture. Sraffa’s disciples founded the neo-Ricardian school based on this framework, but I think it’s possible, and productive, to reintegrate Marx, as Shaikh argued in his 1982 paper Neo-Ricardian economics: a wealth of algebra, a poverty of theory. I believe Ian Wright’s work takes important steps in that direction.

This framework is also used by researchers such as Jason Hickel (the good one) to calculate the flow of labor-hours in the global economy. For example, they found that:

In 2021, the economies of the global North net-appropriated 826 billion hours of embodied labor from the global South, across all skill levels and sectors.

So there are real-life applications of this knowledge. It may also have uses in economic planning.

This isn’t a comprehensive economic theory, nor is it a universal model for Marx’s political economy. Much is left out, such as money and finance. The focus is on production, and the equilibrium prices that follow from it.

The two big quantities this framework determines are:

• The physical gross product (q): the tonnes of iron, gallons of milk, number of automobiles, etc. that must be produced to satisfy final demand.
• The natural prices, or prices of production (p): prices that allow for reproduction of goods and of the involved classes.

I don’t claim this is the only framework for determining prices and understanding value. Alternatives like the temporal single-system interpretation (TSSI) exist, but this is the one I’m most familiar with and able to present in detail.

I’m also not an expert in economics, I’ve tried my best to learn what I can but I don’t have formal training in it (maybe that's a good thing though), and so have blind spots and may be making some mistakes. I’m still learning, but I wanted to share with others what I think (?) I may know and open it to others for criticism and improvement.

Before diving into the math, let’s go over some important caveats.

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On Static Analysis

Before getting into these details, I want to again note that the below framework is not a dynamic description of the economy or of the law of value, hence changes in supply and demand do not come into play. Others in the original thread, as well as my comment discuss the dynamic side of the law of value.

Instead, this comment-chain will discuss the static part of the law of value through the input-output framework. There are a few things to note about this.

1. No Laws of Motion

We aren’t introducing any “laws of motion” for the economy. This framework states how prices, profits, wages, and other quantities must relate to each other. It doesn’t predict what those quantities are, only how they must fit together given whatever they are.

Think of this as a skeleton: no muscles, no motion, just structure.

But perhaps other theories can make use of this skeleton and flesh it out.

2. But Static Doesn’t Mean Still

We should discuss what is meant by a “static model” since the economy, and especially capitalism, is never truly still. When analyzing the static model there are two (that I’m familiar with) interpretations. Both involve thinking of the static model as representing an “equilibrium” state of sorts.

• One method views the equilibrium state as an existing long-term state the economy can be and treats fluctuations from this equilibrium as “mistakes” or brief perturbations.
• Another view, closer to the classical economists and Marx, sees the equilibrium as turbulent gravitation around some regulating center. This is like an attractor in an inherently dynamic and always moving system.

We’ll use the latter interpretation. It does not mean the economy is at equilibrium, but instead that its long-term dynamics are regulated by such a center. This center can also shift over time, especially with economic growth. We will ignore economic growth, though, in this post to make the analysis of this center as simple as possible.

Shaikh uses the term gravitational center, and describes this process as turbulent gravitation. From Shaikh’s Capitalism: Competition, Conflict, and Crisis (2016):

The conventional notion [of equilibrium] assumes that a variable somehow arrives at, and stays at, some balance point… The classical notion of equilibrium is quite different. Average balance is thought to be achieved only through recurrent and offsetting imbalances. Exact balance is a transient phenomenon because any given variable constantly overshoots and undershoots its gravitational center.

(A visual example from Shaikh (2016) of the actual trajectory of economic states x, vs the gravitational center x*. It is the latter we are studying here.)

In nonlinear dynamics an attractor is a state (or set of states) a system tends to evolve toward. But, just like in turbulent gravitation, the real trajectory may never equal the attractor. The path may orbit periodically or behave chaotically. Even if the trajectory never reaches the attractor, though, the attractor can still characterize the evolution of the system and reveal features about the dynamic system itself. Also, frankly, it is easier to analyze the attractor than the exact trajectories.

It is this gravitational center, or attractor, that we are analyzing here, and it must be kept in mind that this center is not the same as the actual trajectory of the economy. It isn’t the actual system, but it still regulates the dynamics so it is worth studying.

3. Macro Without Micro

This framework is explicitly a macro model and doesn’t investigate the micro level of firms or individuals. There is a tendency to think that one must start at the micro level to explain the macro. But 150 years of physics tells us otherwise.

We had thermodynamics before atomic theory. And statistical mechanics, the theory that bridges the micro-states of molecules to the macro-states of thermodynamics, rests on a completely false notion of atoms acting like billiard balls. Yet it still works as a theory. It doesn’t require us to understand the quantum mechanical nature of atoms for us to say something about the macro-world we live in.

This is an example of universality in complexity science: multiple micro-level models can result in the same macro laws. So a macro-level theory can be valid without micro-foundations, or be founded on abstract micro-foundations as long as they are “good enough”. Also, macro-level data or models can’t be used to prove micro-level assumptions because of this multiplicity when moving from micro to macro.

Even more, recent research on emergence shows cases where macro processes are closed from the micro-level, reinforcing the value of starting at the macro-level especially when that’s where your data lives. This doesn’t mean micro-models are useless. They’re valuable for things like policy and full simulations (like with agent-based models). But they’re not required to build a working macro theory.

In essence, you don’t need all micro-level information in order to have a working model of the macro-level. You don’t need to model every molecule of gas to have a climate model. You don’t need to understand DNA molecular physics to do ecology. You don’t need quantum theory to talk about galaxies. You don’t need a perfect micro-level model of individuals to say something about society.

4. Distributions vs. Averages

Some critics, such as Farjoun and Machover in Laws of Chaos (1983), argue that we should model distributions of prices and profits, not just average values. I’m sympathetic to this, but I don’t think it undermines the macro model. In fact, I think the macro model can strengthen their approach.

Macro variables, such as mean price or mean profit rates, can serve as constraints which help solve for the most likely distribution of those quantities (see the maximum entropy approach). This is analogous to how Jaynes used information theory to reformulate statistical mechanics using thermodynamic constraints to find micro distributions.

So discarding the macro-model because it's not micro-distributional is, I think, a mistake.

5. Pre-Institutional vs. Institutional

A final point about the framework discussed here is that of institutions. Pasinetti distinguishes studying economics at the “institutional” (or natural) and the “pre-institutional” level.

The institutional level includes actual markets, firms, governments, etc. The real-world mechanisms that enforce constraints and drive change. The actual economy as it manifests.

The pre-institutional level focuses on constraints: how much labor is required, what output levels are sustainable, etc. These are like the walls of a building: they don’t change quickly, but they shape what’s possible inside.

Real institutions cause dynamism, while the pre-institutional level is analogous to studying the above regulating center. The pre-institutional level doesn’t discuss how the center is met, it simply states the constraints that may exist for the system to reproduce itself.

As Ian Wright puts it

The natural stage reveals the fundamental constraints that any economic system must satisfy, whereas the institutional stage identifies how these constraints manifest in specific institutional setups. The natural constraints are analogous to the interior of a building in which we live. The building doesn’t change. And its interior constrains the possible spaces we might occupy. Nonetheless, very different institutions may be housed by it.

Marx hints at these pre-institutional constraints in his letter to Kugelmann:

Every child knows a nation which ceased to work, I will not say for a year, but even for a few weeks, would perish. Every child knows, too, that the masses of products corresponding to the different needs require different and quantitatively determined masses of the total labour of society. That this necessity of the distribution of social labour in definite proportions cannot possibly be done away with by a particular form of social production but can only change the mode of its appearance , is self-evident. No natural laws can be done away with.

We’ll take a pre-institutional lens here and remain neutral about the specific institutions (e.g., whether labor is commodified or whether markets exist). Some institutional assumptions will sneak in when discussing exploitation or money, but I’ll try to keep things as general as possible. The cut may not be as clean as I am suggesting here.

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Some Final Assumptions and Caveats

We should be upfront about some core assumptions before diving into the math:

Assumptions

1. Capital is circulating capital. Fixed capital (e.g., machines) is ignored or assumed constant. Some models include it, but there’s no standard method I’m aware of.

2. Linear economies of scale Inputs scale linearly outputs. So doubling output means doubling all inputs, including labor.

For assumption 1: All material inputs are used up each cycle. Fixed capital doesn’t directly appear. You can rationalize this by assuming:

• (a) Machine lifespans are long enough to ignore depreciation, or
• (b) Depreciation can be treated as a predictable, scalable input — folded into circulating capital.

Whatever helps you sleep at night, either way we will ignore fixed capital at this level of analysis.

For assumption 2: This can be justified as:

• (a) A first-order approximation that serves as only the stepping stone for a better theory, or
• (b) A valid assumption in a small region of “economic space” near the attractor, where the system behaves approximately linearly.

Either way, this is linear production theory - its limitations must be acknowledged.

Finally, there is no time in this model. That’s intentional. We’re analyzing a non-growing equilibrium - the attractor. Time must be reintroduced when we study dynamics, but for now we abstract from it.

Now let’s begin in Part 2 with the simplest toy model: a pure labor economy

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Part 1 | Part 2 | Part 3 | Part 4 | Part 5 | Part 6

Inspired by discussions our Capital Vol. I Chapter 3 discussions on money I decided to finally set down and give this article by Michael Roberts, The Modern Monetary Trick a read. Especially since we had discussion contrasting Marx's theory of money with that of MMT. I thought I'd share the article here, and give a brief rundown of Robert's main points as I understood them.

I am no expert in money, and I'm still developing my own understanding. This is just another article that I'm in dialogue with, but I thought I'd share it as MMT explicitly came up in our comments on Capital.

Article:

• Red Sails Link - The Modern Monetary Trick
• Original Article in Salvage - The Modern Monetary Trick

My comments below:

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Roberts provides a Marxist critique of MMT and its exponents. The most important part of Robert’s critique is that MMT has no theory of value and hence it is incorrect in arguing that a state can run up a deficit without real consequences under any conditions.

Roberts does admit that certain MMT exponents mention real limitations to money-printing, but do not appear to focus on them sufficiently. There is also a comparison of MMT and Marxist understanding of taxes, as well as a brief discussion toward the end about the American Dollar as the international reserve currency.

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MMT has traction among the left because there is a desire for a theory against balanced budgets and austerity. MMT lends support to the fiscal spending funded by central bank money without fear of budget deficits and debt resulting in a crisis.

MMT adds to the Keynesian idea of government boosting demand that a sovereign government cannot run out of its own currency. The argument is that since a state has a monopoly over fixing its unit of account it can print as much money as needed and boost demand. And as long as there is no full employment (no fully utilized resources), this demanded can be boosted.

Michael Roberts critiques MMT and its theorists in this paper in three parts:

• On a theoretical level: Is MMT the correct theory for how money is created?
• On a practical level: Is there no need to worry about deficit spending?
• On a goal level: What problems does MMT hope to solve?

For Roberts, MMT ...

• … is a theory to justify unrestricted government spending to sustain or restore unemployment. But even the job guarantee that some MMT theorists push have jobs below the minimum wage. Even more, there is no push to change the social structure of capitalism. Hence, MMT obscures the social relations of labor exploitation for profit.
• … doesn’t sufficiently analyze the capitalist sector - the lens is reduced to “state” vs “non-state” actors.
• … doesn’t analyze the contradictions in the capitalist mode of production and hence the causes of crises. Their analysis is similar to those of orthodox Keynesians. For Roberts, the crises are caused by the profitability capital accumulation, not by lack of demand or even austerity in public spending as claimed by MMT and/or Keynesianism.
• ... is a US-centric theory in practice. It offers no practical policy for nations that lack sovereignty (either formal or informal)
• …doesn’t have a theory of value. This is Robert’s most important criticism. There isn’t enough of an emphasis on the real productivity capacity of an economy, and the limitations that it brings (though some MMT advocates mention this). Roberts claims that since value is objective, it is not a product of the law or up to the dictates of the state. MMT ignores value and states that since the state can set the standard of price it essentially can also set value. Any increase in money must be backed by productive capacity, i.e. real value, else there will be consequences to deficits.
• MMT is the iteration of using “tricks of circulation” to save Capitalism, similar to views espoused by Proudhon which Marx critiqued in Capital and the Gundrisse.