The Generality Of The Sraffian Analysis Of The Choice Of Technique

Figure 1: Labor Demanded Per Unit Output in a Stationary State

1.0 Introduction

This post illustrates the analysis of the choice of technique in a case in which marginal products cannot be defined, even in the sense of an interval. It is one more example of the falsity of Austrian and vulgar neoclassical teaching. Perhaps this example suggests the possibility of, say, labor demand "functions" that have a certain thickness or cloudiness.

As far as I know, nobody has used an Austrian flow-input point-output technology to make the point about indeterminacy illustrated by this post. Bellino (1993) presents three examples in which a continuously differentiable, smooth wage-rate of profits curve is consistent with multiple technologies, with the resulting non-differentiability of micro-economic production functions. The technology in none of the three is of the structure used here. Bidard (2014) develops tools for an analysis of Austrian production functions that I found key to developing this post.

2.0 A Simple Economy

Consider a simple capitalist economy, composed of workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. The workers are paid a wage, w (in units of bushels corn per person year) out of the harvest. Capitalists obtain the rate of profits, r.

I specify two technologies, in some sense. Each technology consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques. In each technique, a bushel of corn is produced from inputs of:

• l₀ person-years of labor performed in the year of the harvest.
• l₁ person-years of labor performed one year before the harvest-year.
• l₂ person-years of (unassisted) labor performed two years before the harvest-year.

A technology is fully determined here by specifying all possible values of these dated labor inputs.

2.1 First Technology

Let s be, roughly, an element of a set Q, with Q a subset of the real numbers to be fully specified below. A capitalist knows the minimum labor requirements for each technique in this technology, where a technique is indexed by s. These labor inputs are, in obvious notation:

l_1,0(s) = a + b s

l_1,1(s) = c

l_1,2(s) = 1/(s + 1)

where a, b, and c are positive constants, b is less than one, b is the square of a rational number, and:

b - b^1/2 ≤ a,

(In drawing graphs throughout this post, I use values of a, b, and c of 2, 9/16, and 3, respectively.)

2.2 Second (Sekt) Technology

I learned the word "Sekt" from Bidard; maybe he enjoys champagne. Anyways, in this technology, positive labor inputs only occur in the year of the harvest and two years before. The labor inputs one year before the harvest are zero. For convenience, define the following non-negative constants:

A = a - b + B

B = 2b^1/2 + c

C = a - b + D

D = B/2

In this technology, the techniques are indexed by the variable t, where t is from the subset of the real numbers obtained by removing all elements of Q from the real numbers. The labor inputs are:

l_2,0(t) = C + D t

l_2,1(t) = 0

l_2,2(t) = D/(t + 1)

3.0 The Choice of Technique

As usual, I consider a competitive, steady state economy in which capitalists have chosen the cost-minimizing technique, at an exogenously specified wage or rate of profits. For a given wage, the cost of a technique in the first technology is proportional to v₁(r, s):

v₁(r, s) = (1 + r)² l_1,2(s) + (1 + r) l_1,1(s) + l_1,0(s)

The corresponding function, v₂(r, t), for the second technology is:

v₂(r, t) = (1 + r)² l_2,2(t) + l_2,0(t)

The capitalists choose the technique to minimize v₁(r, s) or v₂(r, t), depending on whether such minimization results in an index being selected from Q or not. The wage-rate of profits curve for a given technique in the first technology is:

w₁(r, s) = 1/v₁(r, s)

I hope the corresponding notation for the wage-rate of profits curve for a technique in the second technology is obvious.

The choice of the cost-minimizing technique results in specifying the indices for the technique in the two technologies as functions of the rate of profits:

s(r) = (1/b^1/2)(1 + r) - 1

t(r) = r

After working out this analysis of the choice of technique for the first technology, I could have re-specified the first technology such that the index for the cost-minimizing technique was always equal to the rate of profits, as in the second technology. I worked backwards, in some sense, following Bidard, such that this nice property obtained for the second technology.

The wage-rate of profits frontier is the outer envelope of the wage-rate of profits curves for the techniques. This frontier (Figure 2) can be shown to be:

w(r) = 1/(A + Br)

I do not specify the technology for the frontier. This example has been constructed such that the both technologies have the identical frontier, when they are extended such that the index for the technique can be any real number. Since the technique varies continuously with the rate of profits, no point on the frontier is a switch point. Further, no technique on the frontier appears more than once. So this is not an example of the reswitching of techniques.

Figure 2: The Wage-Rate of Profits Frontier

I conjecture that a continuum of technologies exist with this frontier. Think of each one of these technologies as corresponding to a constant labor input in the first year before the harvest in the interval [0, c]. I guess, given Bidard's paper, this is an obvious idea.

4.0 Labor Inputs

The analysis of the choice of technique allows one to plot labor inputs, given a complete specification of the technology, versus selected variables from the price system. Accordingly, suppose the first technology is specified only when the index for the technique is a rational number. That is, the set Q is the set of rational numbers. And the set from which the index for the technique in the second technology is the set of irrational numbers.

Figure 3 is an attempt to visualize the labor demanded by firms, per unit output, in the harvest year as a function of the rate of profits. Despite appearances, neither curve in this graph is continuous; they both have an infinite number of holes. The upper curve has a countable infinity of gaps, while the lower curve has an uncountably infinite number of holes. Since these curves are discontinuous everywhere, the marginal product of labor in the harvest year is undefined. One might think of the employment for harvesters as leaping up and down between the curves as the rate of profit varies among economies.

Figure 3: Labor Inputs Per Unit Output in Harvest Year

Perhaps this example is an argument for adopting constructive mathematics in economics. But one can still get leaps between technologies with a continuous variation in the rate of profits, given a more straightforward set Q. Anyways, the labor demanded by firms, per unit output, one year before the harvest leaps between zero and the constant c. Figure 4 shows the labor demanded, per unit output, two years before the harvest. These curves are also discontinuous everywhere.

Figure 4: Labor Inputs Per Unit Output Two Years Before Harvest Year

Suppose, as in Austrian and neoclassical capital theory, that the interest rate (that is, supposedly the rate of profits) were a scarcity index for capital. A higher interest rate would indicate the availability of less capital per worker. Consequently, capitalists would supposedly be encouraged, for this sort of Austrian model, to adopt a technique in which more labor is hired during the harvest year and less during succeeding years (for example, two years beforehand). This idea is consistent with, for example, the first technology. Figures 3 and 4 show that, if one focuses solely on the blue lines, at a higher interest rate, firms want to hire less labor in the given year before the harvest and more during the harvest year. But this idea is inconsistent with the possibility of leaping from one technology to another, as illustrated in Figures 3 and 4. So this example provides another logical proof of the incorrectness of Austrian theory.

As a final step, I want to describe how to generate Figure 1, at the top of this post. In any year in a stationary state, some workers will be gathering the harvest, some will be working on preparing for the harvest one year out (except in the case of the sekt technology), and some will be working on preparing for the harvest two years out. So employment, per the unvarying net output, is the sum of l₀, l₁, and l₂. And these labor inputs can be found from a given rate of profits. From the wage-rate of profits frontier, one can calculate the wage for any given rate of profits. Thus, one has the two dimensions needed to draw the curves in Figure 1. And these curves, as usual here, are discontinuous everywhere. One can think of the labor demanded leaping left and right in the figure as the wage varies. So much for textbook teaching about competitive labor markers.

5.0 Conclusion

The above example has demonstrated, once again, the incoherence of vulgar neoclassical theory. If I thought economists cared about the truth or falsity of their claims, I would be puzzled about mainstream teaching about labor markets and about price theory, more generally.

References

• Enrico Bellino (1993). Continuous Switching in Linear Production Models, Manchester School, V. 61, Iss. 2 (June): pp. 185-201.
• Christian Bidard (2014). The Wage Curve in Austrian Models, Centro Sraffa Working Papers n. 3 (June).