🔬 physics

A note on thermodynamics of the production processes

This paper proposes a heuristic thermodynamic model of production that treats goods creation as a complexity-generating process, introducing "substitutive work" and specific technological capital characteristics to derive production functions that unambiguously decompose output growth into contributions from technology and production factors.

Original authors: Vladimir Pokrovskii

Published 2026-04-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Vladimir Pokrovskii

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are running a massive, high-tech bakery. In the old days, to make bread, you needed a baker (Labor) and a lot of sweat and muscle. But today, you have giant, automated ovens and conveyor belts (Capital).

The big question economists have always struggled with is: How does a machine actually "make" value? Does the bread get more valuable just because a machine baked it, or is it still the baker's work?

Vladimir Pokrovskii's paper argues that we've been looking at this wrong. Instead of thinking of an economy as a business transaction, we should think of it as a giant, complex machine governed by the laws of physics (thermodynamics).

Here is the simple breakdown of his ideas using everyday analogies:

1. The Bakery is a "Complex Machine"

In traditional economics, we say Output = Labor + Capital. But Pokrovskii says, "Wait a minute."
Think of your production system (your factory or bakery) as a living organism or a complex machine.

Labor (L): This is the human effort. The baker pushing the dough.
Capital (K): This is the equipment. The ovens, the mixers, the robots.
Substitutive Work (P): This is the paper's "secret sauce." It's the energy that runs the machines. It's the electricity, the coal, or the wind that makes the robot arm move instead of the baker's arm.

The Analogy: Imagine you are trying to move a heavy rock.

Labor: You pushing the rock with your hands.
Capital: A giant lever you built.
Substitutive Work: The person (or wind) actually pushing the lever.
The paper argues that the "value" of the rock moving comes from the push, whether that push comes from a human hand or a machine powered by electricity.

2. The "Complexity" of Value

The paper suggests that "value" isn't just a price tag; it's a measure of complexity.
When you turn raw flour, water, and yeast into a perfect loaf of bread, you are taking chaotic, messy ingredients and organizing them into something useful. In physics, this is called reducing "entropy" (disorder).

The Goal: The economy is a machine that takes messy raw materials and turns them into organized, useful things.
The Cost: To do this, you need energy. That energy comes from two places: Human Sweat or Machine Power.

3. The Magic of "Substitutive Work"

This is the most important part. The author introduces a new concept called Substitutive Work (P).

Old View: Machines replace humans.
New View: Machines are just tools that allow us to swap human energy for external energy (like electricity or oil).

The Metaphor: Think of a car.

If you walk, you use your own energy (Labor).
If you drive a car, you use the car's engine (Capital) powered by gas (Substitutive Work).
The car didn't "create" the movement; the gas did. But the car (Capital) is the tool that lets the gas do the work. The paper says we need to count the "gas" (Substitutive Work) as a major ingredient in creating value, not just the car or the driver.

4. The Four Ways to Look at the Recipe

The author shows that you can write the "recipe" for economic growth in four different ways, and they all lead to the same result. It's like describing a cake:

By Ingredients: How much flour (Labor) and sugar (Machine Energy) did you use?
By the Pan: How big is the oven (Capital)?
By the Chef: How hard did the baker work?
By the Result: How big is the cake?

The paper proves that if you know the technology (how efficient your oven is), you can predict exactly how much cake you'll get based on how much energy you put in.

5. Why This Matters (The "Aha!" Moment)

Why should a regular person care?

It explains why machines make us richer: It's not magic. It's because machines allow us to use massive amounts of external energy (wind, sun, coal) to do the work that used to require human muscle.
It stops the guessing game: Economists often guess how much "technology" helps growth (the "Solow Residual"). This paper says, "No, we can calculate it exactly." If we know how much energy our machines use and how much labor they replace, we can predict economic growth without guessing.
The "Price" of Machines: The paper suggests that the true cost of a machine isn't just its price tag. It's the cost of the energy it consumes to replace human labor.

The Bottom Line

The economy is a giant thermodynamic engine.

Value is created by organizing chaos into order.
Labor is one way to provide the energy for this.
Machines are the tools that let us swap Labor for External Energy (like electricity).

By understanding that machines are just "energy converters," we can finally understand how technology creates wealth, not just by "being smart," but by doing the heavy lifting that humans used to do. The paper gives us a new, physics-based rulebook for how the economy actually works.

1. Problem Statement

The paper addresses a fundamental puzzle in economic growth theory: how labor is replaced by production equipment (capital).

The Gap: Conventional economic theories (e.g., Solow, Blanchard & Fisher) model output ( $Y$ ) as a function of labor ( $L$ ) and capital ( $K$ ), i.e., $Y = Y(L, K)$ . However, these models often treat the substitution mechanism as an abstract mathematical relationship or rely on exogenous technical progress (the "Solow residual") without explaining the physical mechanism of how machines replace human effort.
The Core Question: How can the creation of value (complexity) be physically understood when human labor is substituted by the work of machines powered by external energy sources? The author argues that existing empirical formulas lack appropriate technological characteristics of equipment and that economic theory needs a firmer physical foundation.

2. Methodology

The author employs a thermodynamic approach, treating the production system not as a black box, but as an open thermodynamic system composed of interacting particles organized in complex, non-equilibrium structures.

System Definition:
- Output ( $Y$ ): Defined as the "complexity" created, interpreted as the diminution of entropy in the human environment. It is measured by market value but rooted in physical energy.
- Inputs: The system requires three distinct factors:
  1. Labor ( $L$ ): Human effort and work.
  2. Substitutive Work ( $P$ ): The work performed by production equipment using external energy (wind, coal, electricity).
  3. Physical Capital ( $K$ ): The stock of equipment (tools/machines) that facilitates the transformation of raw materials.
Technological Characteristics: The author introduces two dimensionless coefficients to describe the quality of capital:
- Labor Requirement ( $\lambda$ ): $\lambda = \frac{K}{L} \frac{dL}{dK}$ (Labor required per unit of capital).
- Energy Requirement ( $\varepsilon$ ): $\varepsilon = \frac{K}{P} \frac{dP}{dK}$ (Energy required per unit of capital).
Derivation: By applying thermodynamic principles (specifically regarding non-equilibrium states and internal variables), the author derives a production function where output is a function of the active work inputs ( $L$ and $P$ ), which are themselves constrained by the passive capital stock ( $K$ ) and its technological coefficients.

3. Key Contributions

The paper makes several theoretical and structural contributions to economic modeling:

Introduction of "Substitutive Work" ( $P$ ): The author formally defines $P$ as a distinct production factor representing the work done by machines. This separates the capacity of capital ( $K$ ) from the work it performs ( $P$ ).
Derivation of Four Production Function Forms: The paper derives four mathematically equivalent formulations of the production function based on thermodynamic first principles:
1. Function of Active Factors: $Y = Y_0 \frac{L}{L_0} \left( \frac{L_0}{L} \frac{P}{P_0} \right)^\alpha$
2. Function of Labor and Capital: $Y = Y_0 \frac{L}{L_0} \left( \frac{L_0}{L} \left[ \frac{K}{K_0} \right]^\varepsilon \right)^\alpha$ (Where $\alpha$ is the technological index).
3. Function of Capital Only: $Y = Y_0 \frac{K}{K_0}$ (Output is proportional to the value of capital).
4. Function of Labor Only: $Y = Y_0 \left( \frac{L}{L_0} \right)^{1/\lambda}$
Reinterpretation of Capital Productivity: The paper argues that capital ( $K$ ) is a passive factor. It does not create value itself; rather, it enables active factors ( $L$ and $P$ ) to perform work. The "marginal productivity of capital" is formally derived as a function of the marginal productivities of labor and substitutive work, weighted by technological coefficients.
Decomposition of Growth: The model allows for an unambiguous decomposition of output growth into:
- Extensive Growth: Driven by increasing the quantity of factors ( $L$ and $P$ ) with constant technology.
- Intensive Growth: Driven by changes in technological characteristics ( $\lambda$ , $\varepsilon$ , and $\alpha$ ), which alter the efficiency of substitution.

4. Results

The Law of Substitution: The paper confirms that labor is replaced not by "capital" in the abstract, but by the work of production equipment powered by external energy.
Proportionality: The model suggests a direct proportionality between output and the monetary value of capital ( $Y \propto K$ ), a relationship supported by historical observations but often overlooked in complex econometric models.
Technological Index ( $\alpha$ ): The index $\alpha = \frac{1-\lambda}{\varepsilon - \lambda}$ determines the degree of substitution between labor and energy. If $\alpha$ changes, it indicates a shift in the production system's structure (technological progress).
Validation: The author notes that Form (10) ( $Y \propto K$ ) and Form (9) have been tested against US historical data (Pokrovskii, 2003; 2018) without contradictions, provided the technological coefficients are accounted for.

5. Significance

Foundational Rigor: The paper attempts to place the theory of economic production on a "firmer thermodynamic footing," moving away from arbitrary parameters and exogenous shocks toward physical laws governing energy and complexity.
Clarifying the Role of Energy: It resolves the debate on whether energy is a production factor by defining "substitutive work" as the mechanism through which external energy replaces human labor.
Policy Implications: The decomposition formulas (14–17) provide a clear framework for policymakers. To increase output, one can either:
1. Increase direct labor expenses.
2. Improve technology to increase the efficiency of substitutive work (lowering $\lambda$ ).
Modeling Utility: By introducing specific technological characteristics ( $\lambda$ and $\varepsilon$ ) as internal parameters, the model expands the ability to analyze production processes without relying on the "Solow residual" as a catch-all for unexplained growth.

In summary, Pokrovskii proposes that economic production is a thermodynamic process where value is created by reducing entropy through the combined efforts of human labor and machine work (substitutive work), facilitated by capital equipment. This framework offers a physically grounded alternative to standard neoclassical production functions.