**1.0 Introduction**
This post presents a model of a steady state with a constant rate of growth in which:

- Total wages and total profits grow at same rate.
- Neutral technical change increases the productivity of labor in all industries.
- The wage per hour increases with productivity.
- Each worker continues to consume the same quantity of produced commodities.
- But each worker takes advantage of increased productivity to work less hours per year.

In these times, when concerns about global warning are so important, one would also want to see
a suggestion of a reduced ecological footprint. So this model of a steady state is only
semi-idyllic.

I do not consider anything in the
mathematical model
below to be original. I outline it
to raise the question whether such a growth path is possible under capitalism. The
model demonstrates logical consistency, but cannot demonstrate that details abstracted
from in the model would prevent its realization.

**2.0 The Model**
Consider a closed economy with no foreign trade. Industries are grouped into two great departments. In
Department I, firms produce means of production, also known as capital goods. The output of
Department I is called ‘steel’ and measured in tons. In Department II, firms produce means of consumption,
also known as consumer goods. The output of Department II is called ‘corn’, measured in bushels.
Both steel and corn are produced from inputs of steel and labor.

Constant coefficients of production (Table 1) are assumed to characterize production in each year. All capital
is circulating capital. Long-lived machines, natural resources, and joint production are abstracted
from in this model. Free competition is assumed. Labor is advanced, and wages are paid out of the net output
at the end of the year. Workers are assumed to spend all of their wages on means of consumption. Profits are
saved at a constant proportion, *s*.

**Table 1: Constant Coefficients of Production**
**Parameter** | **Definition** | **Units** |

*a*_{0, 1}(*t*) | Labor required as input per ton steel produced in year *t*. | Person-Hrs per Ton |

*a*_{1, 1} | Steel services required as input per ton steel produced. | Tons per Ton |

*a*_{0, 2}(*t*) | Labor required as input per bushel corn produced in year *t*. | Person-Hrs per Bushel |

*a*_{1, 2} | Steel services required as input per bushel corn produced. | Tons per Bushel |

Suppose coefficients of production for steel inputs are constant through time but labor coefficients exhibit a growth in
labor productivity of 100 ρ percent:

*a*_{0, j}(*t* + 1) = (1 - ρ) *a*_{0, j}(*t*), *j* = 1, 2

Let *X*_{i}(*t*), *i* = 1, 2; represent the physical output produced in each department
in year *t* and available at the end of the year.
Furthermore, suppose the price of steel, *p*, and the rate of profits, *r*, are constant. Let outputs
from each of the two departments grow at a constant rate of 100 *g* percent:

*X*_{i}(*t* + 1) = (1 + *g*) *X*_{i}(*t*), *i* = 1, 2

Certain quantity equations follow from these assumptions. The quantity of capital goods added each year
must equal the capital goods remaining after reproducing those used up in producing total output, in both
departments:

*g* [*a*_{1,1} *X*_{1}(*t*) + *a*_{1,2} *X*_{2}(*t*)]
= *X*_{1}(*t*) - [*a*_{1,1} *X*_{1}(*t*) + *a*_{1,2} *X*_{2}(*t*)]

The person-years of labor employed relates to labor coefficients and gross outputs:

*L*(*t*) = *a*_{0, 1}(*t*) *X*_{1}(*t*) + *a*_{0, 2}(*t*) *X*_{2}(*t*)

Price equations are:

*p* *a*_{1, 1} (1 + *r*) + *a*_{0, 1} *w*(*t*) = *p*

*p* *a*_{1, 2} (1 + *r*) + *a*_{0, 2} *w*(*t*) = 1

These equations embody the use of a bushel corn as numerate. *w*(*t*) is the wage
per person-hour, paid out at the end of the year out of the surplus.

These assumptions and parameters are enough to depict Table 2. The column labeled "Constant capital" shows the value of
advanced capital goods, taking the output of Department II as the numeraire. The column labeled "Variable Capital" depicts
the wages paid out of revenues available at the end of the year. The surplus is what remains for the capitalists.

**Table 2: A Tableau Economique**
| **Constant** Capital | **Variable** Capital | **Surplus** | **Output** |

I | *p* *a*_{1,1} *X*_{1}(*t*) | *w*(*t*) *a*_{0,1} *X*_{1}(*t*) | *p* *a*_{1,1} *X*_{1}(*t*) *r* | *p* *X*_{1}(*t*) |

II | *p* *a*_{1,2} *X*_{2}(*t*) | *w*(*t*) *a*_{0,2} *X*_{2}(*t*) | *p* *a*_{1,2} *X*_{2}(*t*) *r* | *X*_{2}(*t*) |

I make one further assumption. Workers spend what they get, and capitalists save a constant ration, *s*, of their profits.
With these assumptions, one can calculate the bushels corn that the workers and capitalists in Department I want to purchase,
at the end of each year, from Department II. Likewise, one can calculate the numeraire value of the steel that capitalists
in Department II want to purchase from Depart I. Along a steady state, these quantities must be in balance:

[*a*_{0, 1}(*t*) *w*(*t*) + (1 - *s*) *p* *a*_{1, 1} *r*] *X*_{1}(*t*)
= *p* *a*_{1, 2} [1 + *s* *r*] *X*_{2}(*t*)

This completes the specification of this model of expanded reproduction with technical change uniformly increasing the
productivity of labor.

**3.0 The Solution**
Output per labor hour is found by solving the quantity equations:

*X*_{1}(*t*)/*L*(*t*) = *a*_{1, 2} (1 + *g*)/β(*t*, *g*)

*X*_{2}(*t*)/*L*(*t*) = [1 - *a*_{1, 1} (1 + *g*)]/β(*t*, *g*)

where:

β(*t*, *g*) = *a*_{0, 2}(*t*) + [*a*_{0, 1}(*t*) *a*_{1, 2} - *a*_{0, 2}(*t*) *a*_{1, 1}](1 + *g*)

That is:

*X*_{i}(*t*)/*L*(*t*) = [1/(1 - ρ)^{t}] [*X*_{i}(0)/*L*(0)], *i* = 1, 2

The path of employed labor hours falls out as:

*L*(*t*) = (1 - ρ)^{t} (1 + *g*)^{t} *L*(0)

The number of employed person-hours decreases if:

ρ > *g*

The above expresses the condition that the labor inputs needed to produce a unit of output,
in both departments, decrease faster than the rate of growth in both departments.

The price equations are also easily solved. Given a constant rate of profits, the price of steel is constant as well:

*p* = *a*_{0, 1}(0)/β(0, *r*)

The wage per person-hour increases with productivity:

*w*(*t*) = [1 - *a*_{1, 1} (1 + *r*)/β(*t*, *r*) = [1/(1 - ρ)^{t}] *w*(0)

The trade-offs between consumption per worker and the steady-state rate of growth and between the wage and the rate of profits
have the same form.

These solutions can be substituted into the balance equation. It becomes:

[1 - *a*_{1, 1} (1 + *s* *r*)] (1 + *g*) = [1 - *a*_{1, 1} (1 + *s* *r*)] (1 + *s* *r*)

Suppose the rate of profits falls below its maximum (where the workers ‘live on air’) or not all profits are saved.
Then this is a derivation of the "Cambridge equation":

*r* = *g*/*s*

A steady rate of growth, when the workers consume their wage, requires that the rate of profits be the quotient of the
rate of growth and the savings rate out of profits.

**4.0 Demographics and Institutions**
I make some rather arbitrary assumptions about demographics and institutions. Suppose the number of person-years
supplied as labor grows at the postulated rate of growth:

*L*_{S}(*t* + 1) = (1 + *g*) *L*_{S}(*t*)

with *L*_{S}(*t*) measured in person-years. Let the number of hours in a standard
labor-year, α(*t*) decrease at the same constant rate as the growth in productivity:

α(*t* + 1) = (1 - ρ) α(*t*)

The rate at which the total supply of labor-hours increases is easily calculated:

α(*t* + 1) *L*_{S}(*t* + 1) = (1 - ρ) (1 + *g*) α(*t*) *L*_{S}(*t*)

Under these assumptions, the supply of labor-hours grows at the same rate as the demand for labor-hours. Total wages and
total profits increase at the same rate, 100 *g* percent. The wage per worker increases at the same rate as the
standard length of a labor year declines. Thus, workers consume a constant quantity of commodities, but they
take increased productivity in steadily increased free time.

**5.0 Discussion and Conclusions**
What should one postulate about money in this model? One could assume the money supply grows endogenously, along
with commodities. Or, perhaps, the velocity of the circulation of money increases with productivity. A
continuous decrease in the money price of corn is another logical possibility. Perhaps Rosa Luxemburg
was right, and an external source of demand from less developed regions and countries is needed to
support expanded reproduction. Or Kalecki is correct, and military spending by the government will do.

I do not know if this model describes any existing capitalist economy. It does not describe the post-war
golden age. In that time, at least in the United States, workers took increased productivity in increased
consumer goods. (I think the memory of the Great Depression, the occurrence of World War II, and the
existence of the Soviet Union has something to do how this worked out.)
Could any capitalist economy function like this? Somehow, an advertising industry is not encouraging
workers to consume ever more produced commodities, or they ignore such messages. They continually have more
freedom. Yet, they always spend a bit of time under the domination and direction of their employers.
Will the capitalists tolerate this?