Saturday, April 01, 2017

Bifurcations With Variations In The Rate Of Growth

Figure 1: Perversity and Non-Perversity 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 corn-producing 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 Two-Industry Model

For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the sole corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process.

Each technique is represented by a two-element row vector of labor coefficients and a 2x2 Leontief input-output matrix. For example, the vector of labor coefficients for the Beta technique, a0, β, is:

a0, β = [305/494, 1]

The components of the Leontief matrix for the Beta technique, Aβ, are:

a1,1, β = 229/494
a1,2, β = 2
a2,1, β = 3/1976
a2,2, β = 2/5

The labor coefficients for the Alpha technique, a0, α, differ in the first element from those for the Beta technique. The Leontief matrix for the Alpha technique, Aα, 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 open-source 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 steady-state prices that repeat, year after year, as long as firms adopt the same technique. Let a0 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) + a0 w = p

where p is a two-element 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:

a0 w = p [I - (1 + r) A]

Multiply through, on the right, by the inverse of the matrix in square brackets:

a0 [I - (1 + r) A]-1 w = p

Multiply through, again on the right, by e:

a0 [I - (1 + r) A]-1 e w = p e = 1

Both sides of the above equation are scalars. The wage is:

w = 1/{a0 [I - (1 + r) A]-1 e}

The above equation is called the wage-rate of profits curve or, more shortly, the wage curve. Prices of production are:

p = a0 [I - (1 + r) A]-1/{a0 [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 downward-sloping blue and red curves show that, for each technique, a lower steady-state 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 cost-minimizing 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 cost-minimizing. (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 steady-state 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 labor-year employed, produced in a given year. Let y be the column vector of net quantities, per labor-year. 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 person-year, employment with these quantities is unity:

a0 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 re-arranging terms yields:

c e = [I - (1 + g) A] q


c [I - (1 + g) A]-1 e = q

Multiply through on the left by the row vector of labor coefficients:

c a0 [I - (1 + g) A]-1 e = a0 q = 1

Consumption per person-year is:

c = 1/{a0 [I - (1 + g) A]-1 e}

Gross quantities are:

q = [I - (1 + g) A]-1 e/{a0 [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 trade-off, for the two technique, between steady-state 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 person-year. 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) = a0 w + p A r

Multiply both sides by the vector of gross outputs:

p (I - A) q = a0 q w + p A q r


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


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


(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 cost-minimizing. 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 downward-sloping 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 capital-intensive 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 person-year 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 numeraire-unit 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 non-perverse.

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?)

  • 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. 798-800.
  • Nell (1970). A note on Cambridge controversies in capital theory. Journal of Economic Literature V. 8, No. 1 (March): 41-44.

1 comment:

Sturai said...

I share this paper by Rita Strohmeier on treating Human Capital as Fixed Capital.