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Feasible Sets and the Transformation of Values

Jiyuan Lyu

Abstract

This paper proposes a change in perspective on the ``transformation of values'' problem: from ``searching for a single constant solution'' to ``characterizing the allocation space under objective constraints imposed by the physical production network.'' Building an input--output model, we show mathematically that whenever the macroeconomy features a physical surplus, the set of skilled-to-simple labor reduction vectors that can sustain the subsistence floor of the entire labor force forms a bounded value-feasible set . Within this multidimensional region, the classical ``two great macro equalities'' necessarily hold simultaneously for a reasonable range of profit rates. Hence, without violating the physical minimum conditions for reproduction, the law of value and the nominal price system can be made logically consistent.

Feasible Sets and the Transformation of Values

Abstract

This paper proposes a change in perspective on the ``transformation of values'' problem: from ``searching for a single constant solution'' to ``characterizing the allocation space under objective constraints imposed by the physical production network.'' Building an input--output model, we show mathematically that whenever the macroeconomy features a physical surplus, the set of skilled-to-simple labor reduction vectors that can sustain the subsistence floor of the entire labor force forms a bounded value-feasible set . Within this multidimensional region, the classical ``two great macro equalities'' necessarily hold simultaneously for a reasonable range of profit rates. Hence, without violating the physical minimum conditions for reproduction, the law of value and the nominal price system can be made logically consistent.
Paper Structure (44 sections, 12 theorems, 85 equations, 1 figure)

This paper contains 44 sections, 12 theorems, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theoretical Model and Basic Assumptions
  3. Economic structure
  4. Technological and consumption constraints
  5. The boundary of simple reproduction
  6. The Labor-Reproduction Matrix
  7. Construction
  8. Dimensionlessness and price invariance
  9. Network connectivity and spectral properties
  10. Skilled-Labor Reduction
  11. Reduction coefficients
  12. Constructing the value-feasible domain
  13. Properties of the Value-Feasible Domain and a Benchmark Solution
  14. Spectral equivalence for the existence theorem
  15. Boundedness, closedness, and convexity of the feasible region
  16. ...and 29 more sections

Key Result

Proposition 3.1

All elements of $M_0$ are dimensionless; their magnitudes purely reflect nested proportions of labor time.

Figures (1)

  • Figure 1: The value-feasible domain $\Theta^{val}$ and the geometric mapping of the two macro equalities (two-dimensional slice with $c_1=1$)

Theorems & Definitions (34)

  • Definition 2.1: Intermediate input and capital stock matrices
  • Definition 2.2: Fixed-capital depreciation matrix
  • Definition 2.3: Labor and consumption matrices
  • Definition 3.1: Labor-reproduction matrix
  • Proposition 3.1: Dimensionlessness of $M_0$
  • Lemma 3.2: Price invariance of $M_0$
  • Lemma 3.3: Basic spectral properties of $M_0$
  • Definition 4.1: Value Feasible Domain
  • Remark 4.1: Strict feasibility in the interior
  • Theorem 5.1: Existence theorem for reduction coefficients
  • ...and 24 more