Some Unresolved Issues In Multiple Interest Rate Analysis

1.0 Introduction

Come October, as I understand it, the Review of Political Economy will publish, in hardcopy, my article The Choice of Technique with Multiple and Complex Interest Rates. I discuss in this post questions I do not understand.

2.0 Non-Standard Investments and Fixed Capital

Consider a point-input, flow-output model. In the first year, unassisted labor produces a long-lived machine. In successive years, labor and a machine of a specific history are used to produce outputs of a consumption good and a one-year older machine. The efficiency of the machine may vary over the course of its physical lifetime. When the machine should be junked is a choice variable in some economic models.

I am aware that in this, or closely related models, the price of a machine of a specific date can be negative. The total value of outputs at such a year in which the price of the machine of the machine is negative, however, is the difference between the sum of the price of the machine & the consumption good and the price of any inputs, like labor, that are hired in that year. Can one create a numerical example of such a case in which the net value, in a given year, is negative, where that negative value is preceded and followed by years with a positive net value?

If so, this would an example of a non-standard investment. A standard investment is one in which all negative cash flows precede all positive cash flows. In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments create the possibility that all roots of the polynomial used to define the Internal Rate of Return (IRR) are complex. Can one create an example with fixed capital or, more generally, joint production in which this possibility arises?

Does corporate finance theory reach the same conclusions about the economic life of a machine as Sraffian analysis in such a case? Can one express Net Present Value (NPV) as a function combining the difference between the interest rate and each IRR in this case, even though all IRRs are complex? (I call such a function an Osborne expression for the NPV.)

3.0 Generalizing the Composite Interest Rate to the Production of Commodities by Means of Commodities

In my article, I follow Michael Osborne in deriving what he calls a composite interest rate, that combines all roots of the polynomial defining the IRR. I disagree with him, in that I do not think this composite interest rate is useful in analyzing the choice of technique. But we both obtain, in a flow-input, point output model, an equation I find interesting.

This equation states that the difference between the labor commanded by a commodity and the labor embodied in that commodity is the product of the first input of labor per unit output and the composite interest rate. Can you give an intuitive, theoretical explanation of this result? (I am aware that Osborne and Davidson give an explanation, that I can sort of understand when concentrating, in terms of the Austrian average period of production.)

A model of the production of commodities by means of commodities can be approximated by a model of a finite sequence of labor inputs. The model becomes exact as the number of dated labor inputs increases without bound. In the limit, the labor command by each commodity is a finite value. So is the labor embodied. And the quantity for the first labor input decrease to zero. Thus, the composite interest rate increases without bound. How, then, can the concept of the composite interest rate be extended to a model of the production of commodities by means of commodities?

4.0 Further Comments on Multiple Interest Rates with the Production of Commodities by Means of Commodities

In models of the production of commodities by means of commodities, various polynomials arise in which one root is the rate of profits. I have considered, for example, the characteristic equation for a certain matrix related to real wages, labor inputs, and the Leontief input-output matrix associated with a technique of production. Are all roots of such polynomials useful for some analysis? How so?

Luigi Pasinetti, in the context of a theory of Structural Economic Dynamics, has described what he calls the natural system. In the price system associated with the natural system, multiple interest rates arise, one for each produced commodity. Can these multiple interest rates be connected to Osborne's natural multiple interest rates?

5.0 Conclusion

I would not mind reading attempts to answer the above questions.

The Choice Of Technique With Multiple And Complex Interest Rates

My article with the post title is now available on the website for the Review of Political Economy. It will be, I gather, in the October 2017 hardcopy issue. The abstract follows.

Abstract: This article clarifies the relations between internal rates of return (IRR), net present value (NPV), and the analysis of the choice of technique in models of production analyzed during the Cambridge capital controversy. Multiple and possibly complex roots of polynomial equations defining the IRR are considered. An algorithm, using these multiple roots to calculate the NPV, justifies the traditional analysis of reswitching.

Nonstandard Investments as a Challenge for Multiple Interest Rate Analysis?

1.0 Introduction

This post contains some musing on corporate finance and its relation to the theory of production.

2.0 Investments, the NPV, and the IRR

In finance, an investment project or, more shortly, an investment, is a sequence of dated cash flows. Consider an investment in which these cash flows take place at the end of n successive years. Let C_t; t = 0, 1, ..., n - 1; be the cash flow at the end of the tth year here, counting back from the last year in the investment. That is, C_{n - 1} is the cash flow at the end of the first year in the investment, and C₀ is the last cash flow.

The Net Present Value (NPV) of an investment is the sum of discounted cash flows in the investment. Let r be the interest rate used in time time discounting, and suppose all cash flows are discounted to the end of the first year in the investment. Then the NPV of the illustrative investment is:

NPV₀(r) = C_{n - 1} + C_{n - 2}/(1 + r) + ... + C₀/(1 + r)^{n - 1}

If the above expression is multiplied by (1 + r)^{n - 1}, one obtains the NPV of the investment with every cash flow discounted to the last year in the investment:

NPV₁(r) = C_{n - 1}(1 + r)^{n - 1} + C_{n - 2}(1 + r)^{n - 2} + ... + C₀

For the next step, I need some sign conventions. Let a positive cash flow designate revenues, and a negative cash flow be a cost. Suppose, for now, that the (temporally) first cash flow is a cost, that is negative. Then (-1/C_{n - 1}) NPV₁(r) is a polynomial in (1 + r), with unity as the coefficient for the highest-order term. All other terms are real.

Such a polynomial has n - 1 roots. These roots can be real numbers, either negative, zero, or positive. They can be complex. Since all coefficients of the polynomial are real, complex roots enter as conjugate pairs. Roots can be repeating. At any rate, the polynomial can be factored, as follows:

NPV₁(r) = (-C_{n - 1})(r - r₀) (r - r₁)... (r - r_{n - 1})

where r₀, r₁, ..., r_{n - 1} are the roots of the polynomial. Note that the interest rate appears only in terms in which the difference between the interest rate and one root is taken. And all roots appear on the Right Hand Side. I am going to call an specification of NPV with these properties an Osborne expression for NPV.

Suppose, for now, that at least one root is real and non-negative. The Internal Rate of Return (IRR) is the smallest real, non-negative root. For notational convenience, let r₀ be the IRR.

3.0 Standard Investments in Selected Models of Production

A standard investment is one in which all negative cash flows precede all positive cash flows. Is there a theorem that an IRR exists for each standard investment? Perhaps this can be proven by discounting all cash flows to the end of the year in which the last outgoing cash flow occurs. Maybe one needs a clause that the undiscounted sum of the positive cash flows does not fall below the undiscounted sum of the negative cash flows.

At any rate, an Osborne expression for NPV has been calculated for standard investments characterizing two models of production. As I recall it, Osborne (2010) illustrates a more abstract discussion with a point-input, flow-output example. Consider a model in which a machine is first constructed, in a single year, from unassisted labor and land. That machine is then used to produce output over multiple years. Given certain assumptions on the pattern of the efficiency of the machine, this example is of a standard investment, with one initial negative cash flow followed by a finite sequence of positive cash flows.

On the other hand, I have presented an example for a flow-input, point-output model. Techniques of production are represented as finite series of dated labor inputs, with output for sale on the market at a single point in the time. Each technique is characterized by a finite sequence of negative cash flows, followed by a single positive cash flow.

In each of these two examples, the NPV can be represented by an Osborne expression that combines information about all roots of a polynomial. Thus, basing an investment decision on the NPV uses more information than basing it on the IRR, which is a single root of the relevant polynomial.

4.0 Non-standard Investments and Pitfalls of the IRR

In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments can highlight three pitfalls in basing an investment decision on the IRR:

• Multiple IRRs: The polynomial defining the IRR may have more than one real, non-negative root. What is the rationale for picking the smallest?
• Inconsistency in recommendations based on IRR and NPV: The smallest real non-negative root may be positive (suggesting a good investment), with a negative NPV (suggesting a bad investment).
• No IRR: All roots may be complex.

Berk and DeMarzo (2014) present the example in Table 1 as an illustration of the third pitfall. They imagine an author who receives an advance of $750 hundred thousands, sacrifices an income of $500 hundred thousand in each year of writing a book, and, finally, receives a royalty of one million dollars upon publication. The roots of the polynomial defining the NPV are -1.71196 + 0.78662 j, -1.71196 - 0.78662 j, 0.04529 + 0.30308 j, 0.04529 - 0.30308 j. All of these roots are complex; none satisfy the definition of the IRR.

Table 1: A Non-Standard Investment

Year	Revenue
0	750
1	-500
2	-500
3	-500
4	1,000

5.0 Issues for Multiple Interest Rate Analysis

Osborne, in his 2014 book, extends his 2010 analysis of the NPV to consider the first and second pitfall above. Nowhere do I know of is an Osborne expression for the NPV derived for an example in which the third pitfall arises.

The idea that the pitfalls above for the use of the IRR might be a problem for multiple interest rate analysis was suggested to me anonymously. On even hours, I do not see this. Why should I care about how many roots there are in an Osborne expression for the NPV, their sign, or even if they are complex?

On the other hand, I wonder about how non-standard investments relate to the theory of production. I know that an example can be constructed, in which the price of a used machine becomes negative before it becomes positive. Can the varying efficiency of the machine result in a non-standard investment? After all, the cash flow, in such an example of joint production, is the sum of the price of the conventional output of the machine and the price of the one-year older machine. Even when the latter is negative, the sum need not be negative. But, perhaps, it can be in some examples.

Not all techniques in models with joint production, of the production of commodities by means of commodities, can be represented as dated labor flows. I guess one can still talk about NPVs. Can one formulate an algorithm, based on NPVs, for the choice of technique? How would certain annoying possibilities, such as cycling be accounted for? Can one always formulate an Osborne expression for the NPV? Do properties of multiple interest rates have implications for, for example, a truncation rule in a model of fixed capital? Perhaps a non-standard investment, for a fixed capital example and one pitfall noted above, always has a cost-minimizing truncation in which the pitfall does not arise. Or perhaps the opposite is true.

Anyway, I think some issues could support further research relating models of production in economics and finance theory. Maybe one obtains, at least, a translation of terms.

Appendix: Technical Terminology

See body of post for definitions.

• Flow Input, Point Output
• Investment
• Investment Project
• Internal Rate of Return (IRR)
• Net Present Value (NPV)
• Non Standard Investment
• Osborne Expression (for NPV)
• Point-Input, Flow Output model
• Standard Investment

References

• Jonathan Berk and Peter DeMarzo (2014). Corporate Finance, 3rd edition. Boston: Pearson Education
• Michael Osborne (2010). A resolution to the NPV-IRR debate? Quarterly Review of Economics and Finance, V. 50, Iss. 2: 234-239.
• Michael Osborne (2014). Multiple Interest Rate Analysis: Theory and Applications. New York: Palgrave Macmillan
• Robert Vienneau (2016). The choice of technique with multiple and complex interest rates, DRAFT.

Bifurcations Of Roots Of A Characteristic Equation

Figure 1: Rates of Profits for Beta Technique

I have previously considered all roots of a polynomial equation for the rate of profits in a model, of the choice of technique, in which each technique is specified by a finite series of dated labor inputs. One root is the traditional rate of profits, but there are uses for the other roots:

• All roots appear in an equation defining the Net Present Value (NPV) for the technique, given the wage and the rate of profits.
• All roots can be combined in an accounting identity for the difference between labor commanded and labor embodied, given the wage.

I thought it of interest to know whether these non-traditional roots are real or complex, as they vary with the wage. I am considering multiple roots in an attempt to build on and critique Michael Osborne's approach to multiple interest rate analysis.

I also have considered examples of models of the production of commodities by means of commodities, in which at least one commodity is basic, in the sense of Sraffa. And I have attempted to apply or extend my critique of multiple interest rate analysis to these models. The point of this post is to illustrate possibilities on the complex plane for multiple interest rates in these models.

A technique in models of the production of commodities by means of commodities, as least in the case when all capital is circulating capital, is specified as a vector of labor coefficients and a Leontief input-output matrix. In parallel with my approach to techniques specified by a finite sequence of dated labor inputs, consider wages as being advanced - that is, not paid at the end of the year out of the surplus - in such models. Given the wage and the numeraire, one can construct a square matrix in which each coefficient is the sum of the corresponding coefficient in the Leontief input-output matrix and the quantity of the commodity produced by that industry that is advanced to the workers, per unit output produced. I call this matrix the augmented input-output matrix.

A polynomial equation, called the characteristic equation, is solved to find eigenvalues of the augmented input-output matrix. The power of this polynomial is equal to the number of commodities produced by the technique. The number of roots for the polynomial is therefore equal to the number of commodities. A rate of profits corresponds to each root. Assume the Leontief input-output matrix is a Sraffa matrix and that the wage does not exceed a certain maximum. Under these conditions, the Perron-Frobenius theorem picks out the maximum eigenvalue of the augmented input-output matrix. The corresponding rate of profits is non-negative, and the prices of production of these commodities are positive at the given wage. I was not able to find an application for the other, non-traditional rates of profits.

I present a numerical example in this working paper. This is a three-commodity example with two techniques. Figure 1 graphs the three roots, at different level of wages, for the Beta technique in that example.

In a previous blog post, I extend that example such that managers of firms have a choice of process for producing each of the three commodities. As a consequence, a choice among eight techniques arises. And one can draw a graph like Figure 1 for each technique in that example. Figure 2 shows the corresponding graph for the Delta technique.

Figure 2: Rates of Profits for Delta Technique

In Figures 1 and 2, the rate of profits picked out by the Perron-Frobenius theorem and used to draw the wage-rate of profits curve for the technique lies along the line segment on the real axis on the left in the figure. A lower wage corresponds to a higher traditional rate of profits. Thus, points further to the right on this line segment correspond to a lower wage. A wage of zero leads to the right-most point on this line segment. The highest feasible wage corresponds to left-most point, at a rate of profits of zero, on this segment.

Two non-traditional rates of profits arise for the other two solutions of the characteristic equation. They are plotted to the right on the graphs in Figures 1 and 2. When complex, they are complex conjugates. I thought it of interest that, in Figure 2, they are purely real for two non-overlapping, distinct ranges of feasible levels of the wage.

I draw no practical, applied implications from the non-traditional rates of profits. I just think the graphs are curious.

A Reswitching Example in a Model of Oligopoly

The Roots of a Cubic Polynomial Defining Switch Points

I have a draft paper up at SSRN. The abstract:

This paper illustrates, through a numerical example of reswitching under oligopoly, the existence of implications from the Cambridge Capital Controversy for the theory of industrial organization. Oligopoly is modeled by given and persistent ratios in rates of profits among industries, as expressed in a system of equations for prices of production. The numerical example illustrates that this model of oligopoly is a pertubation of free competition. Some comparisons and contrasts are drawn to a model of free competition.

In some sense, this paper shows a somewhat more comprehensive description of value through exogenous distribution than in Sraffa's book. The model can depict capitalists as squabbling over the division of the surplus that their class gets, as well as their struggle against the workers. I'd like to see an example of reswitching or capital reversing in this model, with all (price and real) Wicksell effects as negative in the example in the special case of free competition. I do not see why one cannot arise. Such an example would suggest that "perverse" examples can obtain empirically, even if they are not found in an analysis that presumes one common rate of profits among all industries.

The graph at the top of this post does not appear in the paper. In the model, the ratios of rates of profits among industries are given parameters. A cubic polynomial is defined for a given set of such ratios. Non-negative, real zeroes of that polynomial below a certain maximum define a scale factor for switch points. The location of the zeros varies with the ratios. I happen to be able to solve for the zeros. They are shown in the graph above.

The Production of Commodities and Multiple Interest Rate Analysis

I've rewritten my analysis of the application of multiple interest rate analysis to models of the production of commodities by means of commodities. (This analysis is limited to circulating capital models, in which there exists no land or long-lasting machines.) I like to think this newer paper is more focused than my earlier paper. For example, I do not have an aside, with graphs, about bifurcation theory, as applied to polynomial equations. I also have an example which I think provides more easily visualizable graphs. I still think these papers are better at raising questions than reaching conclusions.

Trivial Application of Multiple Interest Rate Analysis

I should have put the following in my working paper, on Basic Commodities and Multiple Interest Rate Analysis. This would go somewhere after Equation 10.

Let a technique of production be specified by a row vector, a₀, of labor coefficients and a square Leontief input-output matrix, A. The jth labor coefficient, a_0,j, and the jth column, a_.,j, of A represent the process for producing the j commodity when this technique is in use.

Consider a firm producing the jth commodity with this process. Suppose the firm faces prices of inputs and outputs, as represented by the row vector p. Let w be the given wage and r be the given rate of profits. Then the Net Present Value (NPV) for using this process, per unit output of the j commodity is:

NPV_j(r) = p_j - (p a_.,j + w a_0,j)(1 + r)

Let r₁ be the Internal Rate of Return (IRR) for this process. By definition, the NPV, evaluated for the IRR, is zero:

NPV_j(r₁) = 0

As the appendix proves, one can derive:

NPV_j(r) = - (p a_.,j + w a_0,j)(r - r₁)

In words, when an investment project consists of one payout and one expenditure, with the payout coming one period after the expenditure, the Net Present Value of the investment is the additive inverse of the (first) expenditure, accumulated for one period at the difference between the given rate of profits and the Internal Rate of Return for the investment. Notice that NPV is only positive if the rate of profits used for accumulating costs falls below the internal rate of returns.

This is a trivial application of multiple interest rate analysis because it applies when the multiplicity is one. The above formulation of NPV was suggested to me, however, by first considering a non-trivial application.

Appendix

By the definition of the IRR:

r₁ = [p_j/(p a_.,j + w a_0,j)] - 1

Substitute:

- (p a_.,j + w a_0,j)(r - r₁) = - (p a_.,j + w a_0,j)r + p_j - (p a_.,j + w a_0,j)

Or:

- (p a_.,j + w a_0,j)(r - r₁) = -(p a_.,j + w a_0,j)(r + 1) + p_j

Which is to say:

- (p a_.,j + w a_0,j)(r - r₁) = p_j - (p a_.,j + w a_0,j)(1 + r)

But the term on the right is the definition of NPV. So the two expressions for NPV in the main text are equivalent.

Basic Commodities and Multiple Interest Rate Analysis

I have a new working paper on the Social Science Research Network:

Abstract: This paper considers the application of multiple interest rate analysis to a model of the production of commodities by means of commodities. A polynomial, for the characteristic equation of the augmented input-output matrix, is used in defining the rate of profits in such a model. Only one root is found to be economically meaningful. No non-trivial application of multiple interest rate analysis is found in the analysis of the choice of technique. On the other hand, multiple interest rate analysis can be used in defining Net Present Value in an approximate model, in which techniques are represented as finite series of dated labor inputs. The product of the quantity of the first labor input and the composite interest rate approaches, in the limit, the difference between the labor commanded by and the labor embodied in final output in the full model.

I am proud of some observations in this paper. Nevertheless, I think it tries to go in too many directions at once. It is also longer than I like. It may seem, at first glance, to be longer than it is. I have ten graphs scattered throughout.

Michael Osborne cannot deny that I have taken his research seriously. He needs somebody with more academic credibility than me to write on his topic, though.

This is one paper where I would not mind being shown to be wrong. I did not find any use for more than one eigenvalue of what I am calling the augmented input-output matrix. If somebody can find something useful, along the line of multiple interest rate analysis, to say about all eigenvalues, I would be interested to hear of it.

Update: I accidentally first posted without a "not" in the first sentence of the last paragraph. (I normally silently update typographic errors, but that one changes the meaning.)

Bifurcations In Multiple Interest Rate Analysis

Figure 1: Three Trinomials

1.0 Introduction

Typically, in calculating the Internal Rate of Return (IRR), a polynomial function arises. The IRR is the smallest, non-negative rate of profits, as calculated from a root of this function. The other roots are almost always ignored as having no economic meaning.

Michael Osborne, as I understand it, is pursuing a research project of investigating the use of all the roots of such polynomial functions that arise in financial analysis. A polynomial of degree n has n roots in the complex plane. I have noticed that the roots, other than the IRR, for examples that might arise in practice, can vary in whether they are real, repeating, or complex.

Bifurcation analysis, as developed for the study of dynamic systems might therefore have an application in multiple interest rate analysis. (This post is not about a dynamic system. I do not know how many of these results are about the theory of equations, independently of dynamical systems.) On the other hand, Osborne typically presents his analyses in terms of complex numbers. So I am not sure that he need care about these details.

2.0 An Example

Table 1 specifies the technology to be analyzed in this post. This technology produces an output of corn at the end of one specified year. The production of corn requires inputs of flows of labor in each of the three preceding years (and no other inputs). The labor inputs, per unit corn output, are listed in the table.

Table 1: The Technology

Year Before Output	Labor Hired for Each Technique
1	L1 = 0.18 Person-Years
2	L2 = 4.468 Person-Years
3	L3 = 0.527438298 Person-Years

Let a unit of corn be the numeraire. Suppose firms face a wage of w and a rate of profits, r, to be used for time discounting. Wages are assumed to be advanced. That is, workers are paid at the start of the year for each year in which they supply flows of labor. Accumulate all costs to the end of the year in which the harvest occurs. Then the Net Present Value for this technology is:

NPV(r) = 1 - w[L₁(1 + r) + L₂(1 + r)² + L₃(1 + r)³]

The NPV is a third-degree polynomial. The wage can be considered a parameter. Figure 1, above, graphs this polynomial for three specific values of this parameters. In decreasing order, wages are 11/250, 11/500, and 2/250 bushels per person-years for these graphs.

Given the wage, the IRR is the intersection of the appropriate polynomial with the positive real axis in Figure 1. These IRRs are approximately 101.1%, 175.5%, and 329.5%, respectively. Suppose the economy were competitive, in the sense that capitalists can freely invest and disinvest in any industry. No barriers to entry exist. Then, if this technology is actually in use in producing corn and the wage were the independent variable, the rate of profits would tend to the IRR found for the wage. Profits and losses other than those earned at this rate of profits would be competed away.

The above graph suggests that, perhaps, the NPV for all wages intersects in two points, one of which is a local maximum. I do not know if this is so. Nor have I thought about why this might be. I guess it is fairly obvious that the local maximum is always at the same rate of profits. The wage drops out of the equation formed by setting the derivative of the NPV, with respect to the rate of profits, to zero.

I want to focus on the number of crossings of the real axis in the above graph. Figure 2 shows all roots of the polynomial equation defining the NPV. For a maximum wage, the IRR is zero, and it is greater to the right, along the real axis, for a smaller wage. The corresponding real roots, for the maximum wage, are the greatest and least negative rate of profits along the two loci shown in the left half of Figure 2. For smaller wages, these two real roots lie closer together, until around the middle wage used in constructing Figure 1, only one negative, repeated root exists. For any lower wage, the two roots that are not the IRR are complex conjugates. When the wage approaches zero, the workers live on air and all three roots go to (positive or negative) infinity.

Figure 2: Multiple Rates of Profit for The Technique

This post has presented an example for thinking about multiple interest rate analysis. It is mainly a matter of raising questions. I do not know how the mathematics for investigating these questions impacts practical applications of multiple interest rate analysis.

The Choice Of Technique With Multiple And Complex Interest Rates

I have expanded this post into a working paper. The abstract is:

Abstract: This paper clarifies the relationships between Internal Rates of Return, Net Present Value, and the analysis of the choice of technique in models of production analyzed during the Cambridge capital controversy. Multiple and possibly complex roots of polynomial equations defining the IRR are considered. An algorithm, using these multiple roots to calculate the NPV, justifies the traditional analysis of a reswitching example.

Michael Osborne, I hope, should find the working paper more constructive than my post.

(I do not know why, when I delete comments or mark them as spam, they still remain in the upper right.)

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.