
Figure 1: Rates of Profits for Switch Points for Differential Rates of Profits 
1.0 Introduction
Suppose one knows the technology available to firms at a given point in time. That is, one knows the techniques
among which managers of firms choose. And suppose one finds that reswitching cannot occur under this
technology, given prices of production in which the same rate of profits prevails among all industries.
But, perhaps, barriers to entry persist. If one analyzes the choice of technique for the
given technology, under the assumption that prices of production reflect stable (nonunit) ratios
of profits, differing among industries, reswitching may arise for the technology. The numerical
example in this post demonstrates this logical possibility.
The numerical example follows a model of oligopoly I have previously outlined. In some sense, the example is symmetrical to the example in this
draft paper.
That example is of a reswitching example under pure competition, which becomes an example without reswitching
and capital reversing, if the ratio of the rates of profits among industries differs enough.
The example in this post, on the other hand, has no reswitching or capital reversing under pure
competition. But if the ratios of the rates of profits becomes extreme enough, it becomes a reswitching
example.
2.0 Technology
The technology for this example resembles many I have explained in past posts.
Suppose two commodities, iron and corn, are produced in the example economy.
As shown
in Table 1, two processes are known for producing iron, and one cornproducing process is known.
Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column.
All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital.
Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes.
Table 1: The Technology for a TwoIndustry Model
Input  Iron Industry  Corn Industry 
Labor  1  305/494  1 
Iron  1/10  229/494  11/10 
Corn  1/40  3/1976  2/5 
For the economy to be selfreproducing, both iron and corn must be produced each year. Two
techniques of production are available. The Alpha technique consists of the first ironproducing
process and the lone cornproducing process. The Beta technique consists of the remaining
ironproducing process and the cornproducing process.
3.0 Price Equations
The choice of technique is analyzed on the basis of costminimization, with prices of production.
Suppose the Alpha technique is cost minimizing. Then the following system of equalities and
inequalities hold:
[(1/10)p + (1/40)](1 + rs_{1}) + w = p
[(229/494)p + (3/1976)](1 + rs_{1}) + (305/494)w ≥ p
[(11/10)p + (2/5)](1 + rs_{2}) + w = 1
where p is the price of a unit of iron, and w is the wage.
The parameters
s_{1} and s_{2} are given constants, such that rs_{1}
is the rate of profits in iron production and rs_{2} is the rate of profits
in corn production. The quotient s_{1}/s_{2} is the ratio,
in this model, of the rate of profits in iron production to the rate of profits in
corn production. Consider the special case:
s_{1} = s_{2} = 1
This is the case of free competition, with investors having no preference among industries.
In this case, r is the rate of profits. I call r the scale factor for the
rate of profits in the general case where s_{1} and s_{2}
are unequal.
The above system of equations and inequalities embody the assumption that a unit corn
is the numeraire. They also show labor as being advanced and wages as paid out of
the surplus at the end of the period of production. If the second inequality is
an equality, both the Alpha and the Beta techniques are costminimizing; this is
a switch point. The Alpha technique is the unique costminimizing technique if it
is a strict inequality. To create a system expressing that the Beta technique
is costminimizing,
the equality and inequality for iron production are
interchanged.
4.0 Choice of Technique
The above system can be solved, given s_{1}, s_{2},
and the scale factor for the rate of profits. I record the solution for a couple
of special cases, for completeness. Graphs of wage curves and a bifurcation
diagram illustrate that stable (nonunitary) ratios of rates of profits can
change the dynamics of markets.
4.1 Free Competition
Consider the special case of free competition.
The wage curve for the Alpha technique is:
w_{α} = (41  38r + r^{2})/[80(2 + r)]
The price of iron, when the Alpha technique is costminimizing, is:
p_{α} = (5  3r)/[8(2 + r)]
The wage curve for the Beta technique is:
w_{β} = (6,327  9,802r + 3,631r^{2})/[20(1,201 + 213r)]
When the Beta technique is costminimizing, the price of iron is:
p_{β} = [5(147  97r)]/[2(1,201 + 213r)]
Figure 2 graphs the wage curves for the two techniques, under free competition
and a uniform rate of profits among industries.
The wage curves intersect at a single switch point, at
a rate of profits of, approximately, 8.4%:
r_{switch} = (1/1,301)[799  24 (826^{1/2})]
The wage curve for the Beta technique is on the outer envelope, of the wage curves,
for rates of profits below the switch point. Thus, the Beta technique
is costminimizing for low rates of profits. The Alpha technique is cost minimizing
for feasible rates of profits above the switch point. Around the switch point,
a higher rate of profits is associated with the adoption of a less capitalintensive
technique. Under free competition, this is not a case of capitalreversing.

Figure 2: Wage Curves for Free Competition 
4.2 A Case of Oligopoly
Now, I want to consider a case of oligopoly, in which firms in different
industries are able to ensure longlasting barriers to entry. These
barriers manifest themselves with the following parameter values:
s_{1} = 4/5
s_{2} = 5/4
In this case, the wage curve for the Alpha technique is:
w_{α} = (4,100  4,435r + 100r^{2})/[40(400 + 259r)]
The price of iron, when the Alpha technique is costminimizing, is:
p_{α} = (125  96r)/(400 + 259r)
The wage curve for the Beta technique is:
w_{β} = 8(126,540  195,289r + 72,620r^{2})/[160(24,020 + 9,447r)]
The price of iron, when the Beta technique is costminimizing, is:
p_{β} = 2(3,675  3,038r)/(24,020 + 9,447r)
Figure 3 graphs the wage curves for the Alpha and Beta techniques, for the
parameter values for this model of oligopoly. This is now an example of
reswitching. The Beta technique is cost minimizing at low and high rates
of profits. The Alpha technique is cost minimizing at intermediate rates.
The switch points are at, approximately, a value of the scale factor for
rates of profits of 12.07% and 77.66%, respectively.

Figure 3: Wage Curves for a Case of Oligopoly 
4.3 A Range of Ratios of Profit Rates
The above example of oligopoly can be generalized. I restrict myself to the case where the parameters expressing
the ratio of rates of profits between industries satisfy:
s_{2} = 1/s_{1}
One can then consider how the shapes and locations of wage curves and switch points vary with continuous
variation in s_{1}/s_{2}. Figure 1, at the top of this post, graphs the
wage at switch points for a range of ratios of rates of profits. Since the Beta technique is costminimizing,
in the graph, at all high feasible wages and low scale factor for the rates of profits, I only graph the
maximum wage for the Beta technique. I do not graph the maximum wage for the Alpha technique.
As the ratio of the rate of profits in the iron industry to rate in the corn industry increases towards unity,
the model changes from a region in which the Beta technique is dominant to a reswitching example to an
example with only a single switch point. As expected, only one switch point exists when the rate of
profits is uniform between industries.
5.0 Conclusion
So I have created and worked through an example where:
 No reswitching or capitalreversing exists under pure competition, with all industries earning the same rate of profits.
 Reswitching and capitalreversing can arise for oligopoly, with persistent differential rates of profits across industries.
No qualitative difference necessarily exists, in the long period theory of prices,
between free competition and imperfections of competition.
Doubtless, all sorts of complications of strategic behavior, asymmetric information, and so on
are empirically important. But it seems confused to blame the failure of markets to clear
or economic instability on such imperfections.