
Figure 1: Perversity and NonPerversity in the Labor Market Varying with the Rate of Growth 
1.0 Introduction
I have been considering how the existence and properties of switch points vary with parameters specifying
numerical examples of models of the production of commodities by means of commodities.
Here
are
some
examples
of
such analyses of structural stability. This post adds to this series.
I consider a change in sign of real Wicksell effects to be a bifurcation. In the model in this post,
the steady state rate of growth is an exogenous parameter. So a change of sign
of real Wicksell effects, associated with a variation in the steady state rate of growth, is a
bifurcation.
2.0 Technology
The technology for this example is as usual.
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  2 
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 sole cornproducing process. The Beta technique consists of the remaining
ironproducing process and the cornproducing process.
Each technique is represented by a twoelement row vector of labor coefficients and a 2x2 Leontief inputoutput
matrix. For example, the vector of labor coefficients for the Beta technique, a_{0, β}, is:
a_{0, β} = [305/494, 1]
The components of the Leontief matrix for the Beta technique, A_{β}, are:
a_{1,1, β} = 229/494
a_{1,2, β} = 2
a_{2,1, β} = 3/1976
a_{2,2, β} = 2/5
The labor coefficients for the Alpha technique, a_{0, α},
differ in the first element from those for the Beta technique.
The Leontief matrix for the Alpha technique, A_{α}, differs
from the Leontief matrix for the Beta technique in the first column.
(The mathematics in this post is set out in terms of linear algebra.
I needed to remind myself of how to work out
quantity flows with a positive rate of growth. I solved the example
with Octave, the opensource equivalent of Matlab for the example.
I haven't checked the graphs by also working them out by hand.
You can click on the figures to see them somewhat larger.)
3.0 Prices and the Choice of Technique
Consider steadystate prices that repeat, year after year, as long as firms
adopt the same technique. Let a_{0} and A be the
labor coefficients and the Leontief matrix for that technique. Suppose
labor is advanced and wages are paid out of the surplus at the end of the
year. Then prices satisfy the following system of equations:
p A (1 + r) + a_{0} w = p
where p is a twoelement row vector of prices, w is the
wage, and r is the rate of profits. Let e be a column
vector specifying the commodities constituting the numeraire. Then:
p e = 1
For the numerical example, a bushel corn is the numeraire, and e
is the second column of the identity matrix. I think of the numeraire
as in the proportions in which households consume commodities.
The system of equations for prices of production, including the
equation for the numeraire, has one degree of freedom. Formally,
one can solve for prices and the wage as functions of an
externally given rate of profits. The first equation above
can be rewritten as:
a_{0} w = p [I  (1 + r) A]
Multiply through, on the right, by the inverse of the matrix in square brackets:
a_{0} [I  (1 + r) A]^{1} w = p
Multiply through, again on the right, by e:
a_{0} [I  (1 + r) A]^{1} e w = p e = 1
Both sides of the above equation are scalars. The wage is:
w = 1/{a_{0} [I  (1 + r) A]^{1} e}
The above equation is called the wagerate of profits curve or, more shortly,
the wage curve. Prices of production are:
p = a_{0} [I  (1 + r) A]^{1}/{a_{0} [I  (1 + r) A]^{1} e}
The above two equations solve the price system, in some sense.
Figure 2 plots the wage curves for the example.
The downwardsloping blue and red curves show that, for each technique, a lower steadystate real
wage is associated with a higher rate of profits. The two curves intersect at the two switch
points, at rates of profits of 20% and 80%. For rates of profits between the switch points,
the Alpha technique is costminimizing and its wage curve constitutes the outer envelope
of the wage curves in this region. For feasible rates of profits outside that
region, the Beta technique is costminimizing.
(I talk more about this figure at least twice below.)

Figure 2: Wage Curves also Characterize Tradeoff Between Consumption per Worker and Steady State Rate of Growth 
4.0 Quantities
Suppose the steadystate rate of growth for this economy is 100 g percent. A system of equations, dual
to the price equations, arises for quantity flows.
Let q denote the column vector of gross quantities, per laboryear employed, produced in a given year.
Let y be the column vector of net quantities, per laboryear. Net quantities constitute the
surplus once the (circulating) capital goods advanced at the start of the year,
for a given technique, are replaced:
y = q  A q = (I  A) q
Since quantities are defined per personyear, employment with these quantities is unity:
a_{0} q = 1
By hypothesis, net quantities are the sum of consumption and capital goods to accumulate
at the steady state rate of profits:
y = c e + g A q
Substituting into the first equation in this section and rearranging terms yields:
c e = [I  (1 + g) A] q
Or:
c [I  (1 + g) A]^{1} e = q
Multiply through on the left by the row vector of labor coefficients:
c a_{0} [I  (1 + g) A]^{1} e = a_{0} q = 1
Consumption per personyear is:
c = 1/{a_{0} [I  (1 + g) A]^{1} e}
Gross quantities are:
q = [I  (1 + g) A]^{1} e/{a_{0} [I  (1 + g) A]^{1} e}
Interestingly enough, the relationship between consumption per worker and the rate of growth is identical
to the relationship between the wage and the rate of profits. Thus, Figure 1 is also a graph of the
tradeoff, for the two technique, between steadystate consumption per worker and the rate of growth.
One can think of the abscissa as relabeled the rate of growth and the ordinate as relabeled consumption
per personyear. In the graph, the grey point illustrates consumption per worker at a rate of
growth of 10% for the Beta technique.
The ordinate on this graph is consumption throughout the economy. If the rate of profits exceeds
the rate of growth, both those obtaining income from wages and those obtaining income from
profits will be consuming. When the rates of growth and profits are equal, all profits
are accumulated.
5.0 Some Accounting Identities
The value of capital per worker is:
k = p A q
The value of net income per worker is:
y = p y = p (I  A) q
(I hope the distinction between the scalar y and the vector y
is clear in this notation.)
The value of net income per worker can be expressed in terms of the
sum of income categories.
Rewrite the first equation in Section 3:
p (I  A) = a_{0} w + p A r
Multiply both sides by the vector of gross outputs:
p (I  A) q = a_{0} q w + p A q r
Or:
y = w + k r
In this model,
net income per worker is the sum of wages and profits per worker.
Net income per worker can also be decomposed by how it is spent.
For the third equation in Section 4, multiply both sides by the price vector:
p y = c p e + g p A q
Or:
y = c + g k
Net income per worker is the sum of consumption per worker
and investment per worker.
Equating the two expressions for net income per worker
allows one to derive an interesting graphical feature
of Figure 1. This equation is:
w + r k = c + g k
Or:
(r  g) k = c  w
Or solving for the value of capital per worker:
k = (c  w)/(r  g)
Capital per worker, for a given technique, is
the additive inverse of the slope of two points on the wage
curve for that technique. Figure 1 illustrates for the
Beta technique, with a rate of growth of 10% and
a rate of profits of 80%, as at the upper switch point.
6.0 Real Wicksell Effects
This section and the next presents an analysis confined to prices at the switch point
for a rate of profits of 80%.
For a rate of profits infinitesimally lower than 80%, the Alpha technique is costminimizing.
And for a rate of profits infinitesimally higher, the Beta technique is cost minimizing.
I have explained above how to calculate the value of capital per worker, for the
two techniques, at any given rate of growth.
Abstract from any change in prices of production associated with a change in the
rate of profits. The difference between capital per head for the
Beta technique and capital per head for the Alpha technique, both calculated
at the prices for the switch point, is the change in "real" capital
around the switch point associated with an increase in the rate of profits.
Figure 3 graphs this real Wicksell effect as a function of
the rate of steady state growth.

Figure 3: Variation in Real Wicksell Effect with Steady State Rate of Growth 
Two regions are apparent in Figure 3. The intersection, at the left, of the downwardsloping graph with the axis for the change
in the value of capital per worker shows that the real Wicksell effect is positive, for this switch point, in a stationary state.
Around the given switch point, a higher rate of profits is associated, in a stationary state, with firms wanting to
adopt a more capitalintensive technique. If a greater scarcity of capital caused the rate of profits to rise, so as
to ration the supply of capital, such a logical possibility could not be demonstrated.
The real Wicksell effect, for the switch point at the higher rate of profits, is zero when the rate of growth is equal
to the rate of profits at the other switch points. The value of capital per personyear is the same for the two
techniques, in this case. Consider a line, in Figure 1, connecting the two switch points. It also connects the points
on the wage curve for the Alpha technique for a rate of profits of 80% and a rate of growth of 20%. And the same goes
for the wage curve for the Beta technique.
7.0 Real Wicksell Effects in the Labor Market
A variation in real Wicksell effects with the steady state rate of growth is also manifested in the labor market.
I have echoed above some mathematics which shows that the value of national income is the dot product of a vector
of prices with the vector of net quantity flows. The price vector depends, given the technique,
on the rate of profits at which prices of production are found. The quantity vector depends on the steady state
rate of growth. The reciprocal, (1/y), is the amount of labor firms want to hire, per numeraire unit of
national income, for a given technique. The difference at a switch point between these reciprocals, for the
two techniques, is another way of looking at real Wicksell effects.
Around the switch point at a rate of profits of 80%, a lower wage is associated with firms adopting the Beta
technique. And a higher wage is associated with firms adopting the Alpha technique.
The difference of the above reciprocals, between the Alpha and Beta techniques, is the increase in labor,
per numeraireunit net output, associated with an infinitesimal increase in wages, at the prices
for the switch point. Figure 1 shows this difference, as a function of the steady state
rate of growth, at the switch point with the higher rate of profits in the example.
Figure 1 qualitatively resembles Figure 3. For a stationary state, a higher wage is associated with
firms wanting to employ more labor, per numeraire unit of net output.
This effect is reversed for a high enough steady state rate of growth. The bifurcation, here too,
occurs at the rate of growth for the switch point at 20%.
8.0 Conclusion
This post has illustrated a comparison among steady state growth paths at rates of profits
associated with a switch point. And this switch point is "perverse" from the perspective
of outdated neoclassical theory, at least at a low rate of growth.
But the perversity of this switch point varies with the rate of growth.
In the example, when the rate of growth is between the rate of profits at the
two switch points, the second switch point becomes nonperverse.
And it can go the other way. Real Wicksell effects do not even need to be monotonic.
I need to find an example with at least three commodities, two techniques, and
three switch points. In such an example, the switch point with the largest
rate of profits will have a negative real Wicksell effect for a stationary
state, a positive real Wicksell effect for steady state rates of growth
between the first two switch points, and a negative real Wicksell effect
for higher rates of growth, between the second and third switch points.
(I want to look up Gandolfo (2008) in the light of
past posts.
Can I tell this tale
in terms of increasing returns, instead of exogenous technical change?)
References
 Giancarlo Gandolfo (2008). Comment on "C.E.S. production functions in the light of the Cambridge critique".
Journal of Macroeconomics, V. 30, No. 2 (June): pp. 798800.
 Nell (1970). A note on Cambridge controversies in capital theory. Journal of Economic Literature
V. 8, No. 1 (March): 4144.