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The Falling Rate of Profit under Fixed Capital and Stable Labor Shares

This paper extends the Okishio Theorem by demonstrating that incorporating fixed capital into a multi-sectoral model allows the equilibrium rate of profit to decline despite stable or moderately falling labor shares, driven by a critical wage elasticity threshold and a Prisoner's Dilemma dynamic in technical adoption, a finding empirically supported by Chinese industrial data.

Original authors: Jiyuan Lyu

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jiyuan Lyu

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

The Big Question: Why do profits sometimes fall even when things get better?

For a long time, economists have argued about a famous idea from Karl Marx: that as companies get richer and buy more machines, the overall rate of profit (the return on investment) tends to go down.

However, in 1961, a mathematician named Okishio proved a counter-argument. He showed that if companies adopt new, cheaper technologies, profits should actually go up, not down. This became the "gold standard" for proving Marx was wrong.

This new paper by Jiyuan Lyu says: "Okishio was right, but only if you ignore one huge thing: Fixed Capital."

The author argues that Okishio's math worked for a world of "circulating capital" (like raw materials that get used up immediately). But in the real world, we have fixed capital (machines, factories, software) that lasts for years. When you add these long-lasting machines back into the math, the conclusion flips: Profits can fall even if wages stay stable or grow slowly.

The Core Analogy: The Lemonade Stand vs. The Factory

To understand the paper, let's look at two different business models.

1. The Lemonade Stand (Circulating Capital)

Imagine you run a lemonade stand.

Inputs: Lemons, sugar, and cups. You buy these every morning, make lemonade, sell it, and throw the empty cups away.
The Magic: If you invent a machine that squeezes lemons faster, you save money on labor. Even if the price of lemons goes up a little, your savings on labor usually cover it.
The Result: You make more profit. This is what Okishio proved. If you save on labor, you win.

2. The Factory (Fixed Capital)

Now, imagine you run a massive factory.

Inputs: You still buy raw materials, but you also own a $10 million machine that lasts 10 years.
The Trap: This machine is expensive. Its "cost" isn't just what you paid for it; it's the interest you pay on the loan to buy it, plus the wear and tear (depreciation).
The Twist: The price of this machine depends on the cost of labor to build it.

Here is the "Aha!" moment of the paper:
Imagine you invent a new machine that saves you 20% on labor. Great! But, because you saved labor, the workers demand higher wages (or the economy naturally pushes wages up with productivity).

In the Lemonade Stand, higher wages just cost you a bit more for sugar. You still win.
In the Factory, higher wages make the new machines you buy more expensive to build. Since you have to pay interest on all your existing machines, and those machines are now "worth" more because labor is expensive, your total cost of owning the factory skyrockets.

The paper argues that the "savings" from using less labor are completely eaten up by the "cost" of the expensive machines you are forced to own.

The Three Key Takeaways

1. The "Tipping Point" (The Critical Wage)

The paper calculates a specific "tipping point."

Old View: As long as wages don't rise faster than productivity, profits go up.
New View: There is a lower tipping point. Even if wages rise slower than productivity (which is what usually happens in the real world), profits can still fall if you have a lot of fixed capital (machines).
Analogy: Imagine a seesaw. In the old model, you needed the "Wage" side to be heavier than the "Productivity" side to tip the profit down. In this new model, the "Machine" side is so heavy that even a tiny shift in wages tips the whole thing down.

2. The Prisoner's Dilemma (Why we keep doing it)

This is the most dramatic part. Why do companies keep buying these expensive machines if it hurts everyone's profits?

The Individual View: "If I buy this new machine, I save money on labor. I will make more money than my neighbor. I will win!"
The Group View: "If everyone buys the machine, labor costs go up for everyone, and the machines become too expensive to maintain. Everyone loses."
The Result: It's a Prisoner's Dilemma. Every single company acts rationally to save money, but because they all do it at the same time, the whole system crashes. They are trapped in a race to the bottom.

3. The Real-World Proof (China's Data)

The author didn't just do math; they looked at real data from Chinese factories between 1998 and 2007.

What they found: In industries where machines made up a huge part of the cost (high "fixed capital"), increases in worker productivity did not lead to higher profits.
The dampening effect: The more machines a company had, the less benefit they got from workers becoming more efficient. The "machine tax" ate up all the gains.

Summary: What does this mean for the economy?

This paper suggests that the modern economy is stuck in a trap.

Innovation is a double-edged sword: We invent machines to save labor and boost efficiency.
The cost of ownership: But because we rely so heavily on expensive, long-lasting machines, the cost of owning those machines rises whenever wages rise.
The inevitable fall: Even if workers are paid fairly (or even if their share of the pie shrinks slightly), the sheer weight of our fixed capital (factories, tech, infrastructure) can cause the overall profit rate to fall.

In simple terms: We built an economy that runs on expensive machines. Every time we try to make those machines more efficient, we accidentally make them more expensive to own, which eventually chokes off the profits for everyone. It's like buying a faster car that gets better gas mileage, but the car is so heavy that the engine burns out faster, costing you more in the long run.

1. Problem Statement

The paper addresses a central controversy in political economy: the validity of Marx's Law of the Tendency of the Rate of Profit to Fall (LTRPF) versus the Okishio Theorem.

The Conflict: Marx argued that capital accumulation and rising organic composition of capital depress the profit rate. Okishio (1961) refuted this using a multi-sectoral input-output model, proving that if real wages are constant, cost-reducing technical progress must raise the equilibrium rate of profit.
The Gap: Existing critiques of Okishio often rely on disequilibrium dynamics (Shaikh) or temporal valuation (TSSI), which depart from standard equilibrium methods. Furthermore, previous attempts to reconcile a falling profit rate with stable wages (e.g., Skillman, 1997) required a wage elasticity ( $\mu$ ) greater than 1, implying a rising labor share. This contradicts Kaldor's Stylized Facts ( $\mu = 1$ ) and recent empirical evidence of a stable or declining labor share.
The Question: Can the rate of profit fall in a general equilibrium setting with fixed capital, even when the labor share is stable ( $\mu = 1$ ) or moderately declining ( $\mu < 1$ )?

2. Methodology

The author constructs a multi-sectoral linear production model incorporating fixed capital and endogenous wages, extending the frameworks of Sraffa (1960) and Roemer (1981).

Model Structure:
- Sectors: $n+1$ sectors (1 capital good sector, $n$ consumption/intermediate sectors).
- Capital: Distinguishes between circulating capital (consumed in one period) and fixed capital (depreciated over time with rate $\delta$ ). Fixed capital is treated as a form of joint production.
- Prices: Determined by zero-profit conditions where the price of output covers circulating capital, wages, and the annualized user cost of fixed capital (profit charge + depreciation).
- Wage Mechanism: Real wages ( $w$ $w$ ) adjust to average labor productivity ( $\bar{Y}$ $\overset{ˉ}{Y}$ ) with elasticity $\mu$ $μ$ : $w = w_0 (\bar{Y}/\bar{Y}_0)^\mu$ $w = w_{0} (\overset{ˉ}{Y} / \overset{ˉ}{Y}_{0})^{μ}$ .
  - $\mu = 0$ : Fixed real wage (Okishio premise).
  - $\mu = 1$ : Kaldor's Stylized Facts (constant labor share).
  - $\mu < 1$ : Declining labor share.
Technical Change: Modeled as a binary choice between an old technique $(l, k)$ and a new technique $(\lambda l, \kappa k)$ , where $\lambda < 1$ (labor-saving) and $\kappa > 1$ (capital-deepening).
Analytical Approach:
- Derives closed-form conditions for the equilibrium rate of profit ( $r$ ) using a symmetric input-output matrix (later generalized to asymmetric networks).
- Utilizes Game Theory to analyze the adoption decision of individual firms versus the collective outcome.
- Conducts Numerical Simulations and Empirical Regression Analysis using Chinese Industrial Enterprise data (1998–2007).

3. Key Contributions

Revisiting Okishio with Fixed Capital: The paper demonstrates that the Okishio Theorem's conclusion (profit rate rises) holds only for pure circulating capital. The introduction of fixed capital fundamentally alters the price system because the user cost of fixed capital depends on the price of capital goods, which is sensitive to wage changes.
Critical Wage Elasticity ( $\hat{\mu}$ ): The author establishes the existence of a critical wage elasticity $\hat{\mu}$ $\overset{μ}{^}$ such that $\hat{\mu} < 1$ .
- If $\mu > \hat{\mu}$ , cost-reducing technical progress leads to a falling equilibrium rate of profit.
- Crucially, since $\hat{\mu} < 1$ , a falling profit rate is possible even under Kaldor's Stylized Facts ( $\mu = 1$ ) or a moderately declining labor share.
Prisoner's Dilemma Structure: The paper proves that technical adoption exhibits a strict Prisoner's Dilemma. Individual firms rationally adopt new techniques to capture technological rents (higher individual profit), but universal adoption leads to a lower equilibrium profit rate for all.
Network Effects: In a general asymmetric network, the direction of profit rate movement depends on a sector's "upstreamness" (position in the supply chain) and the degree of capital deepening.

4. Key Results

A. Theoretical Mechanism (The "Hedging Failure")

In a pure circulating capital model, a wage increase ( $\mu=1$ ) exactly offsets the labor-saving reduction in unit costs. However, with fixed capital:

Direct Effect: Capital deepening ( $\kappa > 1$ ) increases the demand for capital goods.
Transmission Effect: A wage increase raises the price of the capital good ( $p_0$ ) because $p_0$ includes the labor cost of producing the capital good.
Amplification: The higher $p_0$ increases the annualized user cost of the entire fixed capital stock ( $r \cdot p_0 k + \delta \cdot p_0 k$ ).
Result: The cost reduction from labor saving is insufficient to offset the increased capital costs driven by the wage-induced price rise of capital goods. The "hedging" mechanism breaks down, causing the effective total labor content to rise, thereby depressing the profit rate.

B. The Critical Thresholds

Theorem 3.3: There exists a closed-form $\hat{\mu} < 1$ . If $\mu > \hat{\mu}$ , then $r_{new} < r_{old}$ .
Property: As fixed capital intensity ( $k$ ) increases, $\hat{\mu}$ decreases, making a falling profit rate more likely even with lower wage elasticities.
Theorem 4.2 (Prisoner's Dilemma): For $\mu > \hat{\mu}$ , "Adopt" is a strictly dominant strategy for every firm, yet the unique Nash equilibrium results in a lower profit rate for the entire economy.

C. Network Generalization (Section 5)

Proportional Change: If all sectors adopt the new technique, the profit rate falls regardless of network structure (provided fixed capital exists).
Single-Sector Change: The effect depends on Composite Upstreamness ( $\Omega_m$ ).
- Upstream sectors (high intermediate demand) may initially see a profit rise if capital deepening is low.
- Downstream sectors or highly capital-intensive sectors see an immediate profit decline.
- However, economy-wide adoption always leads to a decline.

D. Empirical Evidence (Section 7)

Using Chinese Industrial Enterprise data:

Wage Elasticity: Estimated $\mu$ is concentrated around 0.81–0.90, consistent with the theoretical range where $\mu > \hat{\mu}$ .
Interaction Effect: The regression shows a significant negative coefficient on the interaction between Fixed Capital Share ( $\phi$ ) and Labor Productivity Growth.
- Interpretation: In industries with high fixed capital intensity, the positive impact of productivity growth on the profit rate is significantly dampened (or reversed).
Investment Screening: High fixed-capital industries show stricter screening against high capital deepening, consistent with the theoretical constraint $\kappa < \kappa_{max}$ (beyond which adoption is unprofitable even individually).

5. Significance

Theoretical Resolution: The paper resolves the Okishio controversy without abandoning equilibrium analysis or standard national accounting. It shows that the "refutation" of Marx's law relied on the unrealistic assumption of pure circulating capital.
Empirical Relevance: It provides a theoretical basis for the observed long-term decline in profit rates in advanced economies, even in the absence of a rising labor share. It explains why productivity growth does not automatically translate into higher profitability in capital-intensive sectors.
Policy Implications: The findings suggest that in highly automated (fixed-capital intensive) economies, the "coercive laws of competition" drive firms into a collective trap where technological progress erodes the general rate of profit, potentially leading to stagnation or investment crises unless distributional or demand-side interventions occur.

In summary, Lyu demonstrates that fixed capital creates a structural asymmetry that prevents the profit rate from rising in response to labor-saving innovation, even when wages rise in proportion to productivity. This restores the logical consistency of the falling rate of profit tendency within a standard general equilibrium framework.