**1.0 Introduction**
I would like to illustrate triple switching, in
the corn-tractor model,
with one of my one-dimensional diagrams.
I have a triple-switching example,
from Bertram Schefold, but the wage-rate of profit frontier is not visually striking in it.
Such an example would not be worthy
of a research paper.
But perhaps I could modify a section of my
recent working paper
to submit somewhere. Besides, posing a new problem might motivate me to update my computing technology.

**2.0 Technology**
The corn-tractor model
is a fixed capital model, an adaption of
the Samuelson-Gargenani model.
Labor and tractors are used to produce new tractors. Labor and tractors are also used to produce corn.
Corn is the consumption good and the numeraire. Table 1 shows the coefficients of production for a particular type of
tractor.

**Table 1: Inputs for The Technology**
**INPUTS** | **Industry** |

Tractor | Corn |

Labor | *b* person-years | *beta* person-years |

Tractors (of any age) | *a* tractors | *alpha* tractors |

Corn | 0 | 0 |

OUTPUTS | 1 new tractor | 1 bushel corn |

Tractors last *n* years in the tractor industry and *v* years in the corn industry.
Although not apparent in the table, this is an example of joint production. Every process
for producing a new tractor also produces tractors one year older than the tractors used as inputs, except for the process
using (*n* - 1)-year old tractors as an input. Similarly, every process for
producing corn also produces tractors one year older, except for the process using (*v* - 1)-year old tractors.
Tractors operate with constant efficiency for their physical life, albeit with different efficiencies in
the two industry. As assumed in pure fixed capital models, old tractors cannot be transferred between industries.

With these assumptions, no choice of
the economic life of a machine arises.
The tractor will be used for its full physical life in each industry. Only three coefficients need to be specified for each
type of tractor: *a*, *beta*, and (*alpha* *b*). The last is a matter of scaling, of selecting units
of measure for labor or tractors, I guess. Without loss of generality, one can set *alpha* to unity throughout.

**3.0 A Special Case**
To find a triple-switching example, it is apparently sufficient to set *n* = *v* = 2.
Tractors last for two years in both the tractor and the corn industry.
Eventually, I want to consider two types of tractors. The choice of technique
is a matter of choosing the type of tractor to produce and use.
(I always find it mysterious how Steedman and his co-authors find their examples.
One might think this model was thoroughly analyzed decades ago and did not
have anything new to tell us.)

The remainder of this post specifies the solution, in a stationary state, for this special case.

**4.0 Quantity Flows**
Consider a stationary state in which employment is one person-year, across the four operated processes.
Let *q*_{1} be the number of new tractors produced in each process in the tractor industry.
Let *q*_{2} be the bushels corn produced in each process in the corn industry.
These quantities are as follows:

*q*_{1} = *alpha*/{2 [2 *beta* + (*alpha* *b* - *a* *beta*)]}

*q*_{2} = (2 - *a*)/{2 [2 *beta* + (*alpha* *b* - *a* *beta*)]}

One can check this solution. Total employment, *L*, is:

*L* = 2 *b* *q*_{1} + 2 *beta* *q*_{2} = 1

The number of new tractors produced is (2 *q*_{1}), and the
number of new tractors used in production processes is the sum
of (*a* *q*_{1}) and (*alpha* *q*_{2}).
The new tractors used as inputs replace, at the end of the year, the one-year
old tractors used in each industry. So this is a stationary state with employment
of one person-year.

The gross output of corn is also the net output, since corn is not used as an input in production.
That is, consumption per person-year in a stationary state, *c*, is:

*c* = 2 *q*_{2} = (2 - *a*)/[2 *beta* + (*alpha* *b* - *a* *beta*)]

**5.0 Prices of Production**
The system of equations for prices of production are set out in terms of five price variables:

*p*_{0}: The price of a new tractor.
*p*_{1}: The price of a one-year old tractor used in the tractor industry.
*p*_{2}: The price of a one-year old tractor used in the corn industry.
*w*: The wage, in units of bushels per person-year, paid to the workers at the end of the year.
*r*: The rate of profits, assumed to be the same in each of the four production processes.

Sraffa shows how to eliminate the prices of old tractors from the system. This analysis derives
the price of an annuity. The following variable is convenient in setting out the solution of the price equations:

*denom*(*r*) = [(*alpha* *b* - *a* *beta*) *r*^{2}
+ [*beta* + 2 (*alpha* *b* - *a* *beta*)] *r*
+ 2 *beta* + (*alpha* *b* - *a* *beta*)

The wage, as a function of the rate of profits, is:

*w* = [-*a* *r*^{2} + (1 - 2 *a*) *r* + (2 - *a*)]/*denom*(*r*)

I call the above function the wage curve.
The price of a new tractor, also as a function of the rate of profits, is:

*p*_{0} = *b* (*r* + 2)/*denom*(*r*)

The price of a one-year old tractor is:

*p*_{1} = *p*_{2} = *b* (*r* + 1)/*denom*(*r*)

I think that, in this model, as long as the life of a tractor is the same in
producing tractors and corn, the price of an old tractor of a given age does not
vary between the two industries.

The wage curve is also the tradeoff for consumption per worker and the steady state rate of growth.
Accordingly, the wage at a rate of profits of zero is the same as consumption per worker, *c*,
found as a result of the solution of the quantity equations.

**6.0 Conclusion**
I guess I should create a spreadsheet for this special case, but with a choice of two types of tractors.
My problem is to find a set of six coefficients of production, three for each type of tractor, such that
the two wage curves intersect at three points, with positive rates of profits but below the maximum.
Finding such is probably tedious.
Then, I would like to consider perturbations of the coefficients, maybe exponential decreases with time in labor inputs.
And finally, I would like a diagram of, say the wage, for switch points and the maximum, graphed against time.

**Update 9/9/2024:** Added calculation and comment about price of old tractors.