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

Jiyuan Lyu

Abstract

This paper incorporates fixed capital into a multi-sectoral input-output model to reassess the Okishio Theorem. We establish the existence of a critical wage elasticity strictly less than unity, beyond which cost-reducing technical progress leads to a declining equilibrium rate of profit. This implies that profit rates may fall even under Kaldor's Stylized Facts or a moderately declining labour share, significantly extending the theorem's domain of validity. Game-theoretic analysis reveals a strict Prisoner's Dilemma structure underlying technical adoption. Empirical evidence from Chinese industrial data confirms that fixed capital intensity exerts a significant dampening effect on the profit-enhancing impact of productivity growth.

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

Abstract

This paper incorporates fixed capital into a multi-sectoral input-output model to reassess the Okishio Theorem. We establish the existence of a critical wage elasticity strictly less than unity, beyond which cost-reducing technical progress leads to a declining equilibrium rate of profit. This implies that profit rates may fall even under Kaldor's Stylized Facts or a moderately declining labour share, significantly extending the theorem's domain of validity. Game-theoretic analysis reveals a strict Prisoner's Dilemma structure underlying technical adoption. Empirical evidence from Chinese industrial data confirms that fixed capital intensity exerts a significant dampening effect on the profit-enhancing impact of productivity growth.

Paper Structure

This paper contains 38 sections, 14 theorems, 45 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. The Fixed Capital Model
  3. The Economic Environment
  4. Price Equations and the Determination of the Rate of Profit
  5. Endogenous Wage Determination
  6. Technical Progress and the Movement of the Rate of Profit
  7. Characterisation of Technical Change
  8. The Case of a Fixed Wage
  9. The Case of an Endogenous Wage
  10. The Economic Mechanism
  11. Classification of the Parameter Space
  12. The Prisoner's Dilemma
  13. Technological Rents and the Incentive to Adopt
  14. Game Structure and Equilibrium
  15. Generalisation
  16. ...and 23 more sections

Key Result

Proposition 2.1

Let Assumptions ass:symmetric--ass:competition hold, with $\rho(\mathbf{A})<1$ and $\Delta_\delta(R)>0$. Given a real wage $w>0$ satisfying the productiveness condition $w\,\mathcal{L}_\delta(1)<1$, the equilibrium rate of profit $r>0$ is uniquely determined by $\mathcal{L}_\delta(R)$ is strictly increasing in $R$, so that $r$ is strictly decreasing in $w$. The price of the capital good is $p_0=R

Figures (1)

  • Figure 1: Results of the numerical experiments. (a) The critical curve $\hat{\mu}(\kappa)$ lies strictly below $\mu=1$ throughout. (b) The terminal-state rate of profit is decreasing in $\mu$; at $\mu=\hat{\mu}\approx 0.731$ we have $r^{**}=r_0$, and at $\mu=1$ the rate of profit declines by $4.9\%$. (c) The ratio $r^{**}/r_0$ as a function of $\mu$ for different values of $k$: when $k=0$ the ratio equals unity at $\mu=1$; for all $k>0$ it falls below unity. (d) $\hat{\mu}$ is decreasing in $k$, approaching $1$ as $k\to 0$. (e) Single-sector technical change in the asymmetric economy: the upstream sector exhibits $r^{**}>r_0$ for $\kappa<\bar{\kappa}_1\approx 1.69$; non-upstream sectors show $r^{**}<r_0$ for all $\kappa>1$. (f) Economy-wide proportional technical change: the rate of profit is strictly declining for all $\kappa>1$.

Theorems & Definitions (27)

  • Proposition 2.1: Determination of the Rate of Profit
  • Proposition 2.2: Endogenous Foundation of Kaldor's Stylized Facts
  • Proposition 3.1: Single-Sector Innovation under a Fixed Wage
  • Proposition 3.2: Economy-Wide Innovation under a Fixed Wage
  • Theorem 3.3: Necessary and Sufficient Condition for a Declining Rate of Profit
  • Theorem 3.4: Declining Rate of Profit under Single-Sector Innovation
  • Corollary 3.5: Dichotomy of the Parameter Space
  • Proposition 4.1: Equivalent Condition for Positive Technological Rent
  • Theorem 4.2: Prisoner's Dilemma
  • Lemma 5.1: Criterion for Profit Rate Movement
  • ...and 17 more