Multiple And Complex Internal Rates Of Return

Figure 1: One Real and Two Complex Rates of Profit for Alpha Technique

1.0 Introduction

My intent, in this post, is to refute a few lines in Osborne and Davidson (2016). I want to do this in the spirit of this article, while not denying any valid mathematics. Osborne and Davidson have this to say about the numeric example in Samuelson (1968)¹:

In other words, when [the Internal Rate of Return] shifts, affecting the capital cost, the product of the unorthodox rates (the duration of the adjusted labor inputs) also shifts such that the overall interest-rate-cost-relationship is linear. This linearity implies that, in the context of this model at least, switching between techniques can happen but reswitching cannot because two straight lines cross only once. Moreover, the relationship between capital cost and the composite interest rate is positive, implying that the neoclassical 'simple tale' that lower rates promote more roundabout technology, is valid when the interest rate is broadly defined.

Samuelson's example is well-established, and it is incorrect to draw the above conclusion from the Osborne and Davidson model. They derive an equation which, when no pure economic profits exist, relates the price of a consumer good to its cost when a certain composite rate of profits is applied to dated labor inputs. This equation is a tautology; the capital cost on the Right-Hand Side of this equation cannot take on different values without the price on the Left-Hand Side simultaneously varying. Thus, however intriguing this equation may be, it cannot support Osborne and Davidson's supposed refutation of reswitching.

2.0 A Model

Consider a flow-input, point-output model of production of, for example, corn. For a given technique of production, let L_i, i = 1, ..., n; be the input of labor, measured in person-years, hired i years before the output is produced, for every bushel corn produced. Suppose, for now, that a bushel corn is the numeraire². Let the wage, w, be given (in units of bushels per person-year), and suppose wages are advanced. Define:

R = 1 + r,

where r is the rate of profits. The cost per bushel produced is:

w L₁ R + w L₂ R² + ... + w L_n Rⁿ

Define g(R) as the additive inverse of economic profits per bushel produced:

g(R) = w L₁ R + w L₂ R² + ... + w L_n Rⁿ - 1

Divide through by w L_n to obtain a nth degree polynomial, f(r), with a leading coefficient of unity:

f(R) = Rⁿ + (L_{n - 1}/L_n) R^{n - 1} + ... + (L₁/L_n) R - 1/(w L_n)

The Internal Rate of Return (IRR), when this technique is adopted for producing corn, is a zero of this polynomial.

3.0 A Composite Rate of Profits

A nth degree polynomial has, in general, n zeros. These zeros need not be positive, non-repeating, or even real. For a polynomial with real coefficients, as above, some of the zeros can be complex conjugate pairs. The IRR is the rate of profits, r₁, corresponding to the smallest real zero, R₁, exceeding or equal to unity.

r₁ = R₁ - 1 ≥ 0

The IRR is well-defined only if the wage does not exceed the maximum wage, where the maximum wage is the reciprocal of the sum of dated labor inputs for a bushel corn:

w_max = 1/(L₁ + L₂ + ... + L_n)

Let r₂, r₃, ..., r_n be the other n - 1 zeros of the above polynomial. As I understand it, these zeros, especially any complex ones, are ignored in financial analysis. Notice that these rates of profits are calculated, given the quantities of dated labor inputs and the wage. One cannot consider different rates of profits without varying the wage or vice versa.

For any complex number z, one can calculate a corresponding real number, namely, the magnitude (or absolute value):

|z| = |z_real + j z_imag| = [(z_real)² + (z_imag)²]^1/2

where j is the square root of negative one. (I have been hanging around electrical engineers, who use this notation all the time.) Consider the magnitude of the product of all rates of profits associated with the zeros of the polynomial f(R):

| r₁ r₂ ... r_n| = r₁ |r₂| ... |r_n|

One can think of this magnitude as a certain composite rate of profits. Michael Osborne's research project, as I understand it, is to explore the meaning and use of this composite rate of profits in a wide variety of models.

4.0 A Derivation

One can express any polynomial in terms of its zeros. For f(R), one obtains:

f(R) = (R - R₁)(R - R₂)...(R - R_n)

Or:

f(R) = (r - r₁)(r - r₂)...(r - r_n)

Two equivalent expressions of the polynomial of interest can be equated:

Rⁿ + (L_{n - 1}/L_n) R^{n - 1} + ... + (L₁/L_n) R - 1/(w L_n)
= (r - r₁)(r - r₂)...(r - r_n)

The above equation holds for any rate of profits. In particular, it holds for a rate of profits equal to zero. Thus, one obtains the following identity:

1 + (L_{n - 1}/L_n) + ... + (L₁/L_n) - 1/(w L_n) = (-r₁)(-r₂)...(-r_n)

Some algebraic manipulation yields:

(1/w) = (L₁ + L₂ + ... + L_n) - L_n(-r₁)(-r₂)...(-r_n)

Take the magnitude of both sides. One gets:

(1/w) = (L₁ + L₂ + ... + L_n) + L_nr₁ |r₂| ... |r_n|

The above equation, albeit interesting, is a tautology, expressing the absence of pure economic profits. For a given technique (that is, set of dated labor inputs), one cannot consider independent levels of the two sides of the equation. Osborne and Davidson's mistake is to fail to notice that the tautological nature of the above equation invalidates their use of this equation to say something about the (re)switching of techniques.

The Left Hand Side of the above equation is the cost price of a unit output, in terms of person-years. The Right Hand Side is the sum of two terms. The first is the labor embodied in the production of a commodity. The second term is the first labor input, from the most distant time in the past, costed up at the composite rate of profits. Somehow or other, that composite rate of profits, as Osborne and Davidson note, expresses something about the number of time periods over which that first input of labor is accumulated and the distribution of dated labor inputs over those time periods. The number of time periods is expressed in the number of rates of profit that go into forming the composite rate of profits. I find how the distribution of labor inputs affects the composite rate of profits more obscure³. I also wonder how the composite rate of profits appears for a technique in which a first labor input cannot be found.

5.0 Numerical Example

An example might help clarify. Suppose labor inputs, per bushel corn produced, are as in Table 1.

Table 1: The Technology

Year Before Output	Labor Hired for Each Technique
	Alpha	Beta
1	33 Person-Years	0 Person-Years
2	0 Person-Years	52 Person-Years
3	20 Person-Years	0 Person-Years

5.1 Alpha Technique

The number of time periods, n, for the alpha technique, is three. The polynomial whose zeros are sought is:

f_α(R) = R³ + (33/20)R - 1/(20 w)

The maximum wage is (1/53) bushels per person-years. The above polynomial, not having a term for R², is a particularly simple form of a cubic equation. Nevertheless, I choose not to write explicit algebraic expressions for its zeros. Instead, consider the complex plane, as graphed in Figure 1, above. The traditional rate of profits is on the half of the real axis extending to the right from zero. The other two zeros are on the rays shown extending to the northwest and southwest. When the wage is at its maximum, the traditional rate of profits is zero and the complex rates of profits are at the rightmost points on those rays, as close as they ever come to zero. For wages below the maximum and above zero, the rates of profits are correspondingly further away from the origin. Figure 2, on the other hand, graphs the traditional and composite rates of profits, as functions of the wage.

Figure 2: Rate of Profits and Composite Rate of Profits for Alpha Technique

5.2 Beta Technique

For the beta technique, the number of time periods, n, is two. The polynomial whose zeros are sought is:

f_β(R) = R² - 1/(52 w)

For wages not exceeding 1/52 bushels per person-year, the traditional rate of profits is:

r_{1, β} = 1/(52 w)^1/2 - 1

The other rate of profits is:

r_{2, β} = -1/(52 w)^1/2 - 1

The composite rate of profits is:

r_{1, β} | r_{2, β} | = [1/(52 w)] - 1

The dependence of the composite rate of profits on the wage is clearly visible in the beta technique.

5.3 Cost Minimization

Figure 3 graphs the traditional and composite rate of profits, as a function of the wage. In the traditional analysis, the cost-minimizing technique is found by choosing the technique on the outer envelope for the two curves to the left in the figure. Although I do not what meaning to assign to it, one could also form the outer envelope for the two curves on the right, that is, the composite rate of profits. If the (composite) rate of profits is zero, the technique on the outer envelope is the one that intersects the wage axis furthest to the right. This is the technique with the smallest total of dated labor inputs, that is, the beta technique. The outer envelope for both the traditional and composite rate of profits yield the same conclusion.

Figure 3: Wage-Rate of Profits Curves

If one based the choice of technique on the composite rate of profits, one would find the alpha technique preferable for all composite rate of profits above a small rate. This would be a switching example, not a reswitching example. There would only be one switch point, as shown on the diagram. And, by the traditional analysis, it is indeed a reswitching example, with switch points at r₁ equal to 10% and 50%. I still see no reason to believe otherwise or to accept a non-equivalent model.

6.0 Conclusion

Although I reject Osborne and Davidson's conclusion about reswitching, I find the concept of the composite rate of profits intriguing. I suspect Osborne is more interested in impacting corporate finance, with the Cambridge Capital Controversy being a by-the-way kind of application. I do not see how the composite rate of profit helps with the analysis of the choice of technique. Osborne (2010) uses the composite rate of profits to clarify the relationship between the Internal Rate of Return and Net Present Value. I like that in my previous exposition of the above example, I applied an algorithm in which both IRRs and NPVs are relevant. I have not yet absorbed Osborne's NPV analysis.

Footnotes

1. I have an example with reswitching at more reasonable rates of profits.
2. Osborne and Davidson take a person-year of labor as the numeraire. I do not see anything in this model can depend on which commodity is the numeraire.
3. Osborne and Davidson state that the composite rate of profits describes the weighted-average timing of labor inputs. Unlike this average, the Austrian average period of production was originally meant to be defined without references to prices.

Bibliography

• Micheal Osborne (2010). A resolution to the NPV-IRR debate? Quarterly Review of Economics and Finance, V. 50, Iss. 2 (May): pp. 234-239 (working paper).
• Michael Osborne (2014). Multiple Interest Rate Analysis: Theory and Applications, Palgrave Macmillan [I HAVE NOT READ THIS].
• Michael Osborne and Ian Davidson (2016). The Cambridge capital controversies: contributions from the complex plane, Review of Political Economy, V. 28, No. 2: pp. 251-269.
• Paul Samuelson (1968). A summing up, Quarterly Journal of Economics, V. 80, No. 4: pp. 568-583.