**1.0 Introduction**
In my ROBE article, I consider fluke switch points arising from perturbations of coefficients of
production in the Samuelson-Gargenani model, but in the case with only circulating capital.
An obvious generalization is to consider fixed capital. This generalization is
simplified by restricting oneself to the case in which machines operate with constant
efficiency. Steedman (2020) analyzes this case, and this post is a start on working through
elements of the corn-tractor model he leaves as homework. I do not know how far I will
go in rewriting my paper for this case.

**2.0 Technology for a Technique**
In the model, corn is produced by labor working working with a specified type of tractor.
And that type of tractor is itself produced by labor working with that type of tractor.

Each type of tractor defines a technique, where a technique is specified by six parameters:

*a*: The number of tractors (of a given age) whose services are used for a year in producing a new tractor.
*b*: The person-years of labor needed to work with tractors (of a given age) to produce a new tractor.
*n*: The number of years a tractor lasts when used in producing new tractors.
- α: The number of tractors (of a given age) whose services are used for a year in producing a bushel of corn.
- β: The person-years of labor needed to work with tractors (of a given age) to produce corn.
- ν: The number of years a tractor lasts when used in producing corn.

The notation is Steedman's, borrowed from J. R. Hicks. I see that if I keep this notation, I will have
to drop my usual practice, in honor of Joan Robinson, of using lowercase Greek letters to refer to a technique.

Consider, for a technique, the (*n* + ν)-element row vector of labor coefficients **a**_{0},
the (*n* + ν) x (*n* + ν) matrix **A** of input coefficients, and
the (*n* + ν) x (*n* + ν) matrix **B** of output coefficients. This vector and
these matrices have a block structure:

**a**_{0} = | *b* | *b* (**u**^{T})_{1,n - 2} | *b* | β | β (**u**^{T})_{1,ν - 2} | β |

**A** = | 0 | **0**_{1,n - 2} | 0 | 0 | **0**_{1,ν - 2} | 0 |

*a* | **0**_{1,n - 2} | 0 | α | **0**_{1,ν - 2} | 0 |

**0**_{n - 2, 1} | *a* **I**_{n - 2,n - 2} | **0**_{n - 2, 1} | **0**_{n - 2,1} | **0**_{n - 2,ν - 2} | **0**_{n - 2,1} |

0 | **0**_{1,n - 2} | *a* | 0 | **0**_{1,ν - 2} | 0 |

**0**_{ν - 2,1} | **0**_{ν - 2,n - 2} | **0**_{ν - 2,1} | **0**_{ν - 2,1} | α **I**_{ν - 2,ν - 2} | **0**_{ν - 2,1} |

0 | **0**_{1,n - 2} | 0 | 0 | **0**_{1,ν - 2} | α |

**B** = | 0 | **0**_{1,n - 2} | 0 | 1 | (**u**^{T})_{1,ν - 2} | 1 |

1 | (**u**^{T})_{1,n - 2} | 1 | 0 | **0**_{1,ν - 2} | 0 |

*a* | **0**_{1,n - 2} | 0 | 0 | **0**_{1,ν - 2} | 0 |

**0**_{n - 2,1} | *a* **I**_{n - 2,n - 2} | **0**_{n - 2, 1} | **0**_{n - 2, 1} | **0**_{n - 2,ν - 2} | **0**_{n - 2, 1} |

0 | **0**_{1,n - 2} | 0 | α | **0**_{1,ν - 2} | 0 |

**0**_{ν - 2,1} | **0**_{ν - 2,n - 2} | **0**_{ν - 2,1} | **0**_{ν - 2,1} | α **I**_{ν - 2,ν - 2} | **0**_{ν - 2,1} |

Obviously, HTML defeated me here. **I** is the identity matrix, and **u** is a column unit vector.

Each element of **a**_{0} and each column of **A** and **B** correspond to a process
of production. The first *n* columns constitute the tractor sector, and the remaining ν columns
are the corn sector.
I assume constant returns to scale and that each process requires a year to complete.
*a*_{0, j} is the person-years of labor that enters the *j*th
process per unit-level of operations. The *j*th column of **A** is the inputs
consumed by the process, and the *j*th column of **B** is the outputs.
The first row index is for corn. The first row of **A** is zero, since corn is
not used as an input in any process. The second row index is for new tractors.
The remaining row indices are for old tractors.
Once a tractor is used in tbe production of tractors, it can no longer be used
in producing corn. Likewise, a tractor used in the corn sector cannot be transferred
to the tractor sector.

**3.0 An Annuity**
Consider an annuity *c*_{n}(*r*) bought for a dollar at
the start of a year. This annuity pays out the sum *c*_{n}(*r*)
at the end of the first year, at the end of the second year, and so on through the
end of the *n*th year. This arrangement implicitly specifies an interest rate *r*
which equates the cost and the present value of the payments:

1 = *c*_{n}(*r*)/(1 + *r*) + *c*_{n}(*r*)/[(1 + *r*)^{2}] + ... + *c*_{n}(*r*)/[(1 + *r*)^{n}]

A bit of algebra reveals that the payments for the annuity are given by the following formula:

*c*_{n}(*r*) = *r* (1 + *r*)^{n}/[(1 + *r*)^{n} - 1]

The limit as the interest rate approaches zero can be found by L'Hôpital's rule. It is:

*c*_{n}(0) = 1/*n*

I need these formulas below.

**4.0 The Quantity System**
Now I want to consider a steady state in which the economy grows at a uniform rate of 100 *g* percent.
Let the column vector **q** specify the level of operation of each process. I postulate
that **q** has the following form:

**q**^{T} = [*q*_{1}, *q*_{1}/(1 + *g*), ..., *q*_{1}/(1 + *g*)^{n - 1}, *q*_{2}, *q*_{2}/(1 + *g*), ..., *q*_{2}/(1 + *g*)^{ν - 1}]

where *q*_{1} and *q*_{2} are variables to be determined.
Let **e**_{1} be the first column of the identity matrix. Consumption
in a steady-state is *c*(*g*) **e**_{1}, where:

*c*(*g*) **e**_{1} = [**B** - (1 + *g*) **A**] **q**

Expanding the first element of the column vectors on both sides, one gets:

*c*(*g*) = (1 + *g*) *q*_{2}/*c*_{ν}(*g*)

The second element yields:

0 = {[(1 + *g*)/*c*_{n}(*g*)] - (1 + *g*) *a*} *q*_{1} - (1 + *g*) α *q*_{2}

Or:

0 = [1 - *a* *c*_{n}(*g*)] *q*_{1} - α *c*_{n}(*g*) *q*_{2}

Given *g*, the above is a linear equation in *q*_{1} and *q*_{2}.
New tractors do not enter into consumption. Quantity flows are specified such that one
person-year of labor is employed:

**a**_{0} **q** = 1

Or:

(1 + *g*) *b* *q*_{1}/*c*_{n}(*g*) + (1 + *g*) β *q*_{2}/*c*_{ν}(*g*) = 1

Or:

*b* *c*_{ν}(*g*) *q*_{1} + β *c*_{n}(*g*) *q*_{2} = *c*_{n}(*g*) *c*_{ν}(*g*)/(1 + *g*)

A linear system of two equations in two unknowns, given the rate of growth, has now been derived.

The system is easily solved:

*q*_{1} = α *c*_{n}(*g*) *c*_{ν}(*g*) /{[β + α*b**c*_{ν}(*g*) - *a*β*c*_{n}(*g*)](1 + *g*)}

*q*_{2} = [1 - *a**c*_{n}(*g*)] *c*_{ν}(*g*)/{[β + α*b**c*_{ν}(*g*) - *a*β*c*_{n}(*g*)](1 + *g*)}

Consumption per worker (in units of bushels corn per person-year) is:

*c*(*g*) = [1 - *a**c*_{n}(*g*)]/[β + α*b**c*_{ν}(*g*) - *a*β*c*_{n}(*g*)]

In a comparison of steady states, consumption per worker is higher if the rate of growth is
lower.
The dependence of the denominator on the rate of growth vanishes under the special case in which:

*a* *c*_{n}(*g*)/*b* = α *c*_{ν}(*g*)/β

Somehow, the above says that the organic composition of capital does not vary between the
tractor and the corn sectors. The tradeoff, however, between consumption per worker and the rate of
growth is still not linear.
The maximum rate of growth, *G*, is the smallest non-negative
real solution to:

0 = 1 - *a* *c*_{n}(*G*)

Consumption per worker in a stationary state is:

*c*(0) = [*n* - *a*]ν/[*n*νβ + α*b**n* - *a*βν]

One might use the above to discuss the capital-intensity of a technique. If the technique with one
type of tractor is more capital-intensive than the technique with another type, one would
expect *c*(0) to be higher with the first type.

**5.0 The Price System**
I now consider prices. Let **p** be a row vector of prices, *w* the wage, and *r*
the rate of profits. In matrix form, the price equations are:

**p** **A** (1 + *r*) + *w* **a**_{0} = **p** **B**

A bushel corn is the numeraire:

**p** **e**_{1} = 1

The above consists of a system of (*n* + ν + 1) equations for (*n* + ν + 2)
variables. The system has one degree of freedom. Labor is advanced, and wages are
paid out of the surplus at the end of the year. A tractor of each age and history
has a seperate price.

I now rewrite the price equations for the first time. The price of a bushel cotn is unity,
and *p* represents thd price of a new machine. *p*_{m,j}
is the price of a *j*-year old tractor in the tractor sector. *p*_{c,j}
is the price of a *j*-year old tractor in the corn sector. The *n* equations for the
tractor sector are:

*p* *a* (1 + *r*) + *w* *b* = *p* + *p*_{m,1} *a*

*p*_{m,1} *a* (1 + *r*) + *w* *b* = *p* + *p*_{m,2} *a*

...

*p*_{m,n - 1} *a* (1 + *r*) + *w* *b* = *p*

The ν equations for the corn sector are:

*p* α (1 + *r*) + *w* β = 1 + *p*_{c,1} α

*p*_{c,1} α (1 + *r*) + *w* β = 1 + *p*_{c,2} *a*

...

*p*_{c,ν - 1} α (1 + *r*) + *w* β = 1

Consider the equations for the machine sector. Multiply the first equation by (1 + *r*)^{n - 1},
the second equation by (1 + *r*)^{n - 2}, and so on, until the last equation is multiplied
by (1 + r)^{0}.Sum these equations:

*p* *a* (1 + *r*)^{n} + *w* *b* [1 + (1 + *r*) + ... + (1 + *r*)^{n - 1}] = *p* [1 + (1 + *r*) + ... + (1 + *r*)^{n - 1}]

The prices for old tractors appear on both sides of successive equations with the same coefficient and drop
out. A similiar procedure for the corn sector yields:

*p* α (1 + *r*)^{ν} + *w* β [1 + (1 + *r*) + ... + (1 + *r*)^{ν - 1}] = [1 + (1 + *r*) + ... + (1 + *r*)^{ν - 1}]

So far, this procedure works if tractors do not have constant efficiency. The next step requires that, though.
The price equations become:

*p* *a* *c*_{n}(*r*) + *w* *b* = *p*

*p* α *c*_{ν}(*r*) + *w* β = 1

Fpr both the quantity and the price system, a set of (*n* + ν) equations is
reduced to two equations in which (quantities or prices) of old tractors do not enter.
The charge for a tractor is that of an annuity that pays out for each year of the
tractor's life.

The price system is easily solved. The price of a new tractor is:

*p* = *b*/[β + α*b**c*_{ν}(*r*) - *a*β*c*_{n}(*r*)]

Under the special case of equal organic compositions of capital, the ratio of a price of a new
tractor to a bushel corn is the ratio of direct labor inputs. Presumably, prices are also
proportional to labor values in this special case. The wage curve is:

*w* = [1 - *a**c*_{n}(*r*)]/[β + α*b**c*_{ν}(*r*) - *a*β*c*_{n}(*r*)]

I have already discussed the wage curve under the guise of the tradeoff between consumption
per worker and the rate of growth. The maximum rate of profits *R* is identical to
the maximum rate of growth *G*.

**6.0 Conclusion**
Steedman (2020) avoids writing about almost all of the above or leaves it as an exercise for
the reader. Basically, I have derived Steedman's first five numbered equations.
Some of this is in Chapter 10 of Sraffa (1960).

**References**
- Gargenani, Pierangelo. 1970. Heterogeneous capital, the production function and the theory of distribution.
*Review of Economic Studies* 37 (3): 407-436..
- Samuelson, Paul A. 1962. Parable and realism in capital theory: the surrogate production function.
*Review of Economic Studies* 29 (3): 193-206.
- Steedman, Ian. 2020. Fixed capital in the corn-tractor model.
*Metroeconomica* 71: 49-56.
- Vienneau, Robert L. 2018. Normal forms for switch point patterns.
*Review of Behavioral Economics* 5 (2): 169-195.