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ANNEX TO CHAPTER ONE

An Algebraic Model of Extended Reproduction

1. PARAMETERS OF THE SYSTEM

I shall begin with a broad analysis of the system, linking real wages (and surplus-value rates) with the development rates of the productive forces. Each Department (I for production of means of production E and II for production of consumer goods C) is defined, for each phase, by an equation in value terms, as follows:

Phase 1	Department 1	1e + ah = pe1
	Department 2	1e + bh = qc2
Phase 2	Department 1	1e + aδh = pe1
	Department 2	1e + bρh = qc2
Phase 3	Department 1	1e + aδ2h = pe1
	Department 2	1e + bρ2h = qc2, etc.

The first term of each equation stands for the value of constant capital consumed in the production process, reduced to a physical unit of equipment E, estimated at the unit value e (e₁ ≠ e₂ ≠ e₃, etc.) The second term represents the physical quantity a, b, aδ, bρ, etc., of total direct labor (necessary labor and surplus labor) employed by one unit of E in each Department and each phase. The parameter h measures the value product of one hour of labor (not to be confused with hourly wage). The physical product of each department, p and q respectively, is estimated at its unit value e and c (similarly c₁ ≠ c₂ ≠ c₃, etc.).

The system comprises three pairs of parameters (a, b, p, q, δ, and ρ) and two unknowns (e and c) for each pair of equations that describe one phase. Parameters a and b measure the physical labor intensity in the productive process (their reciprocals are related to the organic compositions), parameters p and q represent the physical product of the productive processes using one unit of equipment E in each Department, and parameter t.

Obviously δ and ρ are less than 1 since technical progress enables us to obtain, with less direct labor, a higher physical product per unit of equipment.

2. DETERMINATION OF UNIT PRICES E AND C

If we assume h = 1, the equations supply the pairs e and c:

etc.

As the first set of equations shows, as we produce the capital equipment from capital equipment and direct labor, the unit prices of e fall from one phase to the next at the rate of growth of productivity in Department I. On the other hand, consumer goods being produced from capital equipment and direct labor, the unit prices c fall at a rate that is a combination of δ and ρ.

3. EQUATIONS OF EXTENDED REPRODUCTION

If the capital equipment E is distributed between Departments I and II in the ratios n₁ and 1-n₁, for phase 1, n₂ and 1–n₂ for the next phase, the equations for the production in value terms are as follows:

K is a neutral factor of proportionality.

The dynamic equilibrium of the extended reproduction requires that two conditions be fulfilled:

1. that the wages distributed for each phase (in both Departments) enable the entire output of consumer goods produced during that phase to be bought;

2. that the surplus-value generated during one phase (in both Departments) makes it possible to purchase the entire output of Department I during the next phase.

(a) Equations of supply/demand of consumer goods:

(b) Equations of supply/demand of equipment:

Nominal Wages S are determined as follows:

And Real Wages S′₁ = S₁/c₁ and S′₂ = S₂/c₂ are:

S′₂ > S′₁ since the numerator remains unchanged while the denominator decreases from Phase 1 to Phase 2.

4. NUMERICAL EXAMPLES

Case 1: Equal organic compositions, equal improvement in productivity in the two Departments.

Case 2: Unequal organic compositions, equal improvement in productivity in the two Departments.

Case 3: Equal organic compositions, unequal improvement in productivity (here δ > ρ).

Case 4: The reverse assumption to the preceding case (δ < ρ).

Case 5: Case 3 tending to be limiting, improvement in productivity being confined to Department I (ρ = 1/2 while δ = 1).

Case 6: Limiting case of 4—improvement in productivity is confined to Department II (δ = 1/2 while ρ = 1).