The Cambridge Equation, Expanded Reproduction, and Markup Pricing: An Example

1.0 Introduction

I have sometimes set out Marx's model of expanded reproduction, only with prices of production instead of labor values. I assume two goods, a capital good and a consumption good, are produced with constant technology. If one assumes workers spend all their wages and capitalists save a constant proportion of profits, one can derive the Cambridge equation in this model.

The Cambridge equation shows that, along a steady state growth path, the economy-wide rate of profits is determined by the ratio of the rate of growth and the saving rate out of profits. Maybe one should not use causal language here. The Cambridge equation is a necessary, consistency condition for smooth reproduction in a capitalist economy.

This post derives the Cambridge equation with markup pricing, in a highly aggregated model of expanded reproduction. I am curious how far this result generalizes. I am thinking of a model in which, say, n capital goods are produced in Department I and m consumer goods are produced in Department II. At this point, I am not thinking of generalizations in which workers save and therefore own some of the capital stock. Nor am I worrying about fixed capital, depreciation, and technical change.

Table 1: Definition of Variables

Variable	Definition
a01	The person-years of labor hired per unit output (e.g., ton steel) in the first sector.
a02	The person-years of labor hired per unit output (e.g., bushel corn) in the second sector.
a11	The capital goods (measured in tons) used up per unit output in the first (steel-producing) sector.
a12	The capital goods (measured in tons) used up per unit output in the second (corn-producing) sector.
p1	The price of a unit output in the first sector.
p2	The price of a unit output in the second sector.
s1	Relative markup in producing steel.
s2	Relative markup in producing corn.
r̂	The scale factor for the rate of profits.
r	The rate of profits.
σ	The savings rate out of profits.
w	The wage, that is, the price of hiring a person-year.
c	Consumption per worker, in units of bushels per person-year.
X1	The number of units (ton steel) produced in the first sector.
X2	The number of units produced (bushels corn) in the second sector.
g	The rate of growth.

2.0 The Model

Certain quantity equations follow from the assumptions. No produced capital goods remain each year after subtracting those used to reproduce the capital goods used up in throughout the economy and those needed to support the given rate of growth:

0 = X₁ - (1 + g)(a₁₁ X₁ + a₁₂ X₂)

Consumption per person year is the output of the second department:

c = X₂

The model economy is scaled such that one person-year is employed:

a₀₁ X₁ + a₀₂ X₂ = 1

I have the usual price equations, with labor advanced:

p₁ a₁₁ (1 + r̂ s₁) + a₀₁ w = p₁

p₁ a₁₂ (1 + r̂ s₂) + a₀₂ w = p₂

The consumption good is the numeraire:

p₂ = 1

As with Marx in volume 2 of Capital, industries are here grouped into two great departments (Table 1). Means of production (also known as capital goods) are produced in Department I, and means of consumption (or consumer goods) are produced in Department II.

Table 2: Value of Outputs by Department and Distribution

Department	Capital	Wages	Profits
I. Capital Goods	a11 X1 p1	a01 X1 w	a11 X1 p1 s2 r̂
II. Consumption Commodities	a12 X2 p1	a02 X2 w	a12 X2 p1 s2 r̂

The overall, economy-wide rate of profits is defined in terms of profits and capital advances, aggregated over both departments:

r = (a₁₁ X₁ p₁ s₂ r̂ + a₁₂ X₂ p₁ s₂ r̂)/(a₁₁ X₁ p₁ + a₁₂ X₂ p₁)

The economy experiences expanded reproduction when it consistently expands each year. In this case, the demand for capital goods from the second department includes the savings of the capitalists receiving profits from that department. Likewise, the demand for consumption goods from the first department excludes the savings of the capitalists in that department. Observing these qualifications, it is easy to mathematically express the condition that the demand for capital goods from the second department match the demand for consumption goods from the first department:

a₀₁ X₁ w + (1 - σ) a₁₁ X₁ p₁ s₂ r̂ = a₁₂ X₂ p₁ + σ a₁₂ X₂ p₁ s₂ r̂

3.0 Some Aspects of The Model Solution

Quantity variables (c, X₁, and X₂) can be found as a function of the rate of growth. Price variables (w, p₁, and p₂) can be found as a function of the scale factor for the rate of profits. These solutions allow one to use the balance equation to find a relation between the scale factor for the rate of profits:

r̂ = (g/σ){1/[s₂ - (1 - g)(s₂ - s₁)a₁₁]}

One can use the above relationship and the solution quantities and prices to find the economy-wide rate of profits:

r = g/σ

Along a path in which the economy steadily expands, the rate of profits must be equal to the quotient of rate of growth and the savings rate out of profits. The rate of profits is dependent on investment and savings decisions, out of the control of the workers. (In a two-class economy in which the workers save at a smaller rate than the capitalists, the Cambridge equation remains valid, with the savings rate in the denominator being that of the capitalists.) It is independent of the technical conditions of the chosen technique, and marginal productivity has nothing to do with it.

4.0 Conclusions

I know that this model can be generalized to hold when any number of consumer goods are produced. I have not yet been able to show the Cambridge equation holds when any number of capital goods are produced.