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