A Model Of Oligopoly

1.0 Introduction

Suppose barriers to entry exist in an economy. Entrepreneurs and capitalists find that they cannot freely enter or exit some industries. And these barriers are manifested by stable ratios of rates of profits among industries. This post presents equations for prices of production under these assumptions.

I suggest that the model presented here fits into the tradition of Old Industrial Organization, as formulated by Joe Bain and Paolo Sylos Labini. As I understand it, Sylos Labini may have once written down equations like these, but never presented them or published them. I suppose this model is also related to work Piero Sraffa published in the 1920s.

2.0 The Model

Consider an economy consisting of n industries. Suppose the rate of profits in the jth industry is (s_j r), where r is the base rate of profits, s_j is positive, and:

s₁ + s₂ + ... + s_n = 1

For simplicity, I limit my attention to a circulating capital model of the production of commodities by means of commodities. For the technique in use, let a_{i, j} be the quantity of the ith commodity used to produce a unit of output in the jth industry. Homogeneous labor is the only unproduced input in each industry. Let a_{0, j} be the person years of labor used to produce a unit output in the jth industry. I assume labor is advanced, and wages are paid out of the surplus at the end of production period, say, a year. Then prices of production, which ensure a smooth reproduction of the economy, satisfy the following system of equations:

(a_{1, 1} p₁ + a_{2, 1} p₂ + ... + a_{n, 1} p_n)(1 + s₁ r) + w a_{0, 1} = p₁

(a_{1, 2} p₁ + a_{2, 2} p₂ + ... + a_{n, 2} p_n)(1 + s₂ r) + w a_{0, 2} = p₂

. . .

(a_{1, n} p₁ + a_{2, n} p₂ + ... + a_{n, n} p_n)(1 + s_n r) + w a_{0, n} = p_n

The coefficients of production, including labor coefficients, and the ratios of the rate of profits are given parameters in the above system of equations. The unknowns are the prices, the wage, and the base rate of profits. Since only relative prices matter in this model, one degree of freedom is eliminated by choosing a numeraire:

p₁ q^*₁ + ... + p_n q^*_n = 1

Since there are n price equations, appending the above equation for the specified numeraire yields a model with (n + 1) equations and (n + 2) unknowns. One degree of freedom remains.

3.0 In Matrix Form

The above model can be expressed more concisely in matrix form. Define:

• I is the identity matrix.
• e is a column vector in which each element is 1.
• S is a diagonal matrix, with s₁, s₂, ..., s_n along the principal diagonal.
• p is a row vector of prices.
• q^* is the column vector representing the numeraire.
• A is the Leontief input-output matrix, representing the technique in use.
• a₀ is the row vector of labor coefficients for the technique.

The model consists of the following equations:

e^T S e = 1

p A (I + r S) + w a₀ = p

p q^* = 1

4.0 Conclusion

One could develop the above model in various directions. For example, one could plot the wage-base rate of profits curve for the technique in use. Of interest to me would be presenting examples of the choice of technique, including reswitching and capital-reversing. The Sraffian critique of neoclassical economics is not confined to the theory of perfect competition.

Update (16 January 2017): I find I have outlined this model before.

Reswitching In An Example Of A One-Commodity Model

Figure 1: The Choice of Technique in a Model with One Commodity

1.0 Introduction

This post presents a reswitching example in a one-good model. The single produced commodity in the example can be used as both a consumption and a capital good. It is produced by expenditure of labor with its services. It lasts for three production periods, and its technical efficiency varies over the course of its lifetime, when used as a capital good.

I do not remember any comparable numeric example in the literature. If I recall Ian Steedman's 1994 article, he gives instructions for constructing a one-good example, but does not present one. Maybe if I reread it now, I will find it clearer. I have previously worked through a reswitching example, with fixed capital, from J. E. Woods. But that is a multi-commodity model. I have also once echoed Sraffa's analysis of depreciation charges, in a case with constant efficiency.

2.0 Technology

This is an example of fixed capital, a kind of joint production. Three production processes are known by the manager of firms, and they each exhibit constant returns to scale. Each process requires a year to complete, and each process produces new widgets. Table 1 shows the inputs for each process, when operated at a unit level, and Table 2 shows the outputs. For example, process I requires inputs of labor and new widgets. The outputs of process I consist of new widgets and the widgets which provided their services throughout the year it is under operation. Those leftover widgets are one year older, though. Consumers consume new widgets during the year following on their purchase. The physical life of widgets, when providing services for production, is three years. Thus, three processes can be operated in production.

Table 1: Inputs for The Technology

Input	Process
	(I)	(II)	(III)
Labor	10	60	13/2
New Widgets	1/3	0	0
One-Year Old Widgets	0	1/3	0
Two-Year Old Widgets	0	0	1/3

Table 1: Outputs for The Technology

Output	Process
	(I)	(II)	(III)
New Widgets	1	7/12	79/20
One-Year Old Widgets	1/3	0	0
Two-Year Old Widgets	0	1/3	0

Firms are not required to operate all three processes. They can truncate the use of widgets after one or two years. Assume free disposal, that is, that discarding widgets does not incur a cost. Under these assumptions, three techniques are available to produce new widgets, as shown in Table 3.

Table 3: Techniques

Technique	Processes
Alpha	I
Beta	I, II
Gamma	I, II, II

3.0 Price Equations

Managers of widget-producing firms choose the technique, that is, the truncation period, on the basis of cost. As usual, consider a competitive economy, in which workers and firms are free to seek out higher wages and profits, respectively. Revenues and costs are calculated on the basis of a set of prices in which workers and firms have no incentive to move out of one process and into another. Workers receive a common wage of w new widgets per person-year. Assume workers are paid out of the surplus at the end of each year. Firms receive a rate of profits of 100 r percent in operating each process in use. For notational convenience, define R:

R = 1 + r

Let p₁ be the price of a one-year old widget and p₂ the price of a two-year old widget.

I confine the systems of price equations for the Alpha and Beta techniques to an appendix. Accordingly, assume the Gamma technique is in use. The wage, prices, and the rate of profits must satisfy a system of three equations:

(1/3)R + 10 w = 1 + (1/3) p₁

(1/3) p₁ R + 60 w = 7/12 + (1/3) p₂

(1/3) p₂ R + (13/2) w = 79/20

Given the rate of profits below some maximum, one can solve for the wage:

w_γ = (60 R² + 35 R + 237 - 20 R³)/(10 (60 R² + 360 R + 39))

The price of two-year old and one-year old widgets fall out:

p_{2, γ} = (237 - 390 w_γ)/(20 R)

p_{1, γ} = (7 + 4 p_{2, γ} - 720 w_γ)/(4 R)

If you feel like it, you can substitute on the Right Hand Sides of the above two equations so as to express prices as functions exclusively of the rate of profits.

4.0 Choice of Technique

I have explained above how to find the wage, as a function of the rate of profits, when the Gamma technique is in use. The wage-rate of profits curves for the Alpha and Beta technique are, respectively:

w_α = (3 - R)/30

w_β = (12 R + 7 - 4 R²)/(120 (R + 6))

Figure 1 graphs all three wage curves. The cost-minimizing technique, at any given rate of profits, is the technique on the outer envelope of the wage curves. The switch points between the Alpha and Gamma techniques are at rates of profits of 10% and 50%. Below 10% and above 50%, the Gamma technique is cost-minimizing. Widgets are used in production processes to the extent of their physical life. Between these rate of profits, the Alpha technique is cost minimizing. The use of widgets, as capital goods, is truncated after one year. For what it is worth, the switch point between the Alpha and Beta techniques, within the outer envelope, is at a rate of profits of 41/24, approximately 171%.

4.1 A Direct Method with Alpha Prices

In the general theory of joint production, an analysis of the choice of technique cannot generally be based on wage-rate of profits curves. Such an analysis does work in this model of fixed capital. But I checked it with a more direct method of analysis.

Suppose the Alpha technique is in use. The prices of one-year old and two-year old widgets are zero. The wage is as found from the system of price equations associated with the Alpha technique. Would it pay to produce new widgets with one-year old or two-year old widgets? Figure 2 shows calculations to determine if supernormal profits can be earned with the Beta or Gamma technique.

Figure 2: Supernormal Profits at Alpha Prices

Since the Alpha technique is in use, its net present value is zero. Extra profits are assumed to have been competed away.

If the Beta technique is operated, no extra profits or losses are earned in operating process I under the Beta technique. In operating process II, the services of old widgets are free, and no revenues are received for the two-year old widgets disposed of at the end of the year. At low rates of profits and high wages, the revenues received for new widgets produced with process II do not cover labor costs. At high rates of profits, the opposite is the case. These prices, when wages are low, signal to firms that they can earn extra profits by extending the truncation period one year.

The analysis for the Gamma technique is more cumbersome. Firms can not adopt process III without also operating process II. Accordingly, the net present value for the Gamma technique is found by accumulating all costs and revenues for all three processes to the end of the third year. In such a weighted sum, the revenues for process I are multiplied by (1 + r)², and the revenues for process II are multiplied by (1 + r). At a rate of profits below 10% and above 50%, firms will want to adopt the Gamma technique and produce with old widgets to the end of their physical life.

4.2 A Direct Method with Beta Prices

I find of interest some complications that arise in applying this direct method with wages and prices, as calculated for the Beta price system. Figure 3 shows the net present value for the processes comprising each of the three techniques. At all rates of profits, these prices signal that firms should extend the truncation period, from two years, to the three years specified by the Gamma technique. This is so, even for rates of profits between 10% and 50%. If firms start at a two-year truncation period, they will only find that they need to truncate to one year after first extending production to three years. (See A.3 in the Appendix for graphs associated with the Gamma price system.)

Figure 3: Supernormal Profits at Beta Prices

When the Beta technique is operated, a price must be assigned to the price of a one-year old widget. In the theory of joint production, prices can be negative when calculated for rates of profits below the maximum. (This is not so for pure circular capital models without joint production.) Figure 4 graphs the price of one-year old widgets, under the system of prices associated with the Beta technique. This price is negative for rates of profits of approximately 171%. Confine your attention to the Alpha and Beta techniques for a second. The negative price of one-year old widgets signals firms that it is profitable to truncate production from two years to one year.

Figure 4: Price of One-Year Widget with Beta Technique

The application of a direct method for comparing costs and revenues for techniques of production confirms, in the context of this model of fixed capital, the results of constructing the outer envelope curve from the wage-rate of profits curves for the techniques.

5.0 Conclusion

The literature on macroeconomics contains many models with aggregate production functions and in which the one produced commodity can be either consumed by households or used as a capital good in further production. And, in many of these models, this capital good depreciates over many periods. These models are one-good models, in the sense of this post. Some mainstream economists ignorantly assert that the Cambridge Capital Controversy was exclusively about problems in aggregating capital goods. Since mainstream economists are aware of aggregation issues, they somehow conclude they are justified in ignoring the controversy. I usually refute this rot by pointing out consequences for microeconomics of the analysis of the choice of technique. This post takes an alternative approach. It examines a highly aggregated model. And issues related to Sraffa effects arise in the one-good model, too.

This example also has a bearing on a misunderstanding common among the Austrian school of economics. Böhm Bawerk, at least, thought of production processes taking a longer amount of time as being more capital intensive and, therefore, more productive, in some sense. Firms are supposedly restricted in how willing they are to temporally extend production processes because of the scarcity of capital, as reflected in the interest rate. If households were less impatient and more willing to save, the interest rate would fall and firms would adopt longer processes. One can find many Austrian school economists (for example, Hayek in the 1930s) rejecting the idea that there exists a meaningful quantitative measure of roundaboutness or the period of production, whether independent of prices or not. But Austrian school economists generally retain a sense that the theory is insightful and somehow qualitatively true. The numeric example challenges this idea. One would think that a truncation of the production process, with the capital good not reaching its physical lifetime, is unambiguously less roundabout. As noted in Figure 1, for one switch point in the example, such a truncation can be associated with a lower interest rate.

Appendix A

I confine various mathematical details to this appendix.

A.1 Alpha Price Equations

If the Alpha technique is in use, the prices of one-year old and two-year old widgets is zero:

p_{1, α} = p_{2, α} = 0

The wage and the rate of profits are related by the coefficients of production for process I:

(1/3)R + 10 w = 1

The wage, under the Alpha technique, can be expressed as a function of the rate of profits, as illustrated in Figure 1.

A.2 Beta Price Equations

If the Beta technique is in use, the price two-year old widgets is zero:

p_{2, β} = 0

A system of two equations arises for the Beta technique:

(1/3)R + 10 w = 1 + (1/3) p₁

(1/3) p₁ R + 60 w = 7/12

The wage as a function of the rate of profits, for the Beta technique, is also illustrated above. The price of one-year old widgets is:

p_{1, β} = R + 30 w_β - 3

A.3 Direct Method with Gamma Prices

This section presents two graphs with wages and the rate of profits found from the system of prices for the Gamma technique. Figure 5 shows the net present value of truncating the use of widgets after one, two, or three years. For all techniques, revenues and costs are accumulated, at the going rate of profits to the end of the last year in which widgets are produced with the technique. The net present value for operating the Alpha technique (truncating after one year) is positive between the switch points at rates of profits of 10% and 50%.

Figure 5: Supernormal Profits at Gamma Prices

Figure 6 shows the prices of one-year old and two-year old widgets for the solution to the Gamma price system. Although not very easy to see in the graph, the price of one-year old widgets is negative at rates of profits between 10% and 50%.

Figure 6: Price of Widgets with Gamma Technique

References

• Ian Steedman. 1994. 'Perverse' Behaviour in a 'One Commodity' Model. Cambridge Journal of Economics, V. 18, No. 3: pp. 299-311.

On Ajit Sinha On Sraffa

Over at the Institute for New Economic Theory (INET), Ajit Sinha discusses the Sraffian revolution. Scott Carter cautions that, in interpreting Sraffa's thought, his archives have barely been touched.

Sinha's article has this blurb, with which I entirely agree:

"The prominence of the debate over 'reswitching' has obscured the importance of Piero Sraffa's profound contribution to economics. It's time to revisit and build on that body of work."

One can agree with the above without following Sinha very far. In analyzing the choice of technique, I often point out more than reswitching. I try to find effects in other markets than the capital markets and go in other directions. Since my motivation for working through these examples is frequently an internal criticism of neoclassical economics, I am frequently willing to assume Constant Returns to Scale and perfect competition, in the sense that firms take prices as given. One might argue that this misses Sraffa's point. Besides one can use 'reswitching' as a synecdoche for such analyses of the choice of technique.

How do I know that Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory was about more than Sraffa effects, as seen in the analysis of the choice of technique? Only the last chapter in the book deals with the choice of technique. (Maybe, earlier chapters on joint production, rent, and fixed capital might have been clearer if they came after this chapter.) Sraffa doesn't present this one-chapter, final part of his book as a climax that all before is leading up to. In fact, he explicitly says, in the first paragraph, that the status of that chapter is somewhat different from the rest of the book:

"Anyone accustomed to think in terms of the equilibrium of demand and supply may be inclined, on reading these pages, to suppose that the argument rests on a tacit assumption of constant returns in all industries. If such a supposition is found helpful, there is no harm in the reader's adopting it as a temporary working hypothesis. In fact, however, no such assumption is made. No changes in output and (at any rate in Parts I and II [Part III presents switches in methods of production - RLV]) no changes in the proportions in which different means of production are used by an industry are considered, so that no question arises as to the variation or constancy of returns. The investigation is concerned exclusively with such properties of an economic system as do not depend on changes in the scale of production or in the proportions of 'factors'."

The analysis of the choice of technique shows that much neoclassical teaching and "practical" applications is humbug. But that does not exhaust Sraffa's point. Turning to the first sentence of the next paragraph in the preface can help:

"This standpoint, which is that of the old classical economists from Adam Smith to Ricardo, has been submerged and forgotten since the advent of the 'marginal' method."

A second major emphasis of Sraffa's scholarship, including his 1960 book, is the rediscovery of the logic of the classical theory of value and distribution. Sraffians can claim to have a theory that can serve as an alternative to neoclassical theory and that is empirically applicable (for example, by Leontief and those aware of the National Income and Product Accounts (NIPA).) This rediscovery provides an external critique of neoclassical theory.

By the way, the development of this external critique provides, for example, Pierangelo Garegnani for a defense of the claim that the analysis of the attraction of market prices to prices of production is building on Sraffa's work. Sraffa's book does not discuss market processes or the classical theory of competition:

"A less one-sided description than cost of production seems therefore required. Such classical terms as 'necessary price', 'natural price' or 'price of production' would meet the case, but value and price have been preferred as being shorter and in the present context (which contains no reference to market prices) no more ambiguous." (PoCbMoC, p. 9)

One could read Sraffa as being able to take many aspects of classical political economy as given, including analyses of market prices. How should ideas that Sraffa explicitly choose to include in his archives, but not publish in his lifetime, influence our interpretation?

None of this gets to Sinha's point. He thinks, as I understand it, that Sraffa offers more than a rediscovery of classical political economy. Sraffa offers innovations in our understanding of prices and distribution, and these innovations can help us better understand actually existing capitalist economies. (Some of these innovations might be Wittgenstein-like in that they allow us to improve by discarding lots of rubbish.) I daresay Scott Carter agrees with that claim, even though he might disagree with details of Sinha's understanding of the Standard Commodity.

Tangency of Wage-Rate of Profits Curves

Figure 1: The Choice of Technique in a Model with One Switch Point

1.0 Introduction

This post presents an example in which vertically-integrated firms producing a consumption good have a choice between two techniques. The wage-rate of profits curves for the techniques have a single switch point, at which they are tangent. I, and, I dare say, most economists who are aware of the illustrated possibility, consider this a fluke, a possibility that cannot be expected to arise in practice.

2.0 Technology

The technology in this example has the structure of Garegnani's generalization of Samuelson's surrogate production function. One commodity, corn, can be produced from inputs of labor and a single capital good. Two processes are known for producing corn, and each process requires a different capital good, called "iron" and "copper". Each capital good is produced, if at all, by a process that requires inputs of labor and that capital good. Each process requires a year to complete, and the services of the capital good fully consume that capital good during the course of the year. No stock of iron or copper remains at the end of the year to carry over into the next year.

Constant Returns to Scale are assumed for each process. Table 1 shows the coefficients of production for the four processes specified by the technology. Each column corresponds to a process. The coefficients of production specify the input of the row commodity that is needed to produce a unit output of the commodity for the column.

Table 1: The Technology for a Three-Industry Model

Input	Iron Industry	Copper Industry	Corn Industry
	(a)	(b)	(c)	(d)
Labor	1	17,328/8,281	1	361/91
Iron	1/2	0	3	0
Copper	0	48/91	0	1
Corn	0	0	0	0

Two techniques are available for producing corn (Table 2). The Alpha technique consists of the process for producing iron and the corn-producing process that requires an input of iron. The Beta technique consists of the copper-producing process and the corn-producing process using services provided by copper.

Table 2: Techniques

Technique	Processes
Alpha	a, c
Beta	b, d

3.0 Prices and the Choice of Technique

The technique, as usual, is chosen by managers of firms to minimize costs. Corn is taken as the numeraire, and wages are paid at the end of the year. Prices of production, in which all extra profits above a common rate have been competed away, are used to calculate costs. For analytical convenience, in this pose I take the rate of profits as given.

Suppose the Alpha technique is chosen. Under the above assumptions, the price of iron and the wage must satisfy the following system of two equations:

(1/2) p_iron (1 + r) + w_α = p_iron

3 p_iron (1 + r) + w_α = 1

A similar system arises for the Beta technique, but as applied to the price of copper and the coefficients of production for the Beta technique.

For a non-negative rate of profits, up to a certain maximum rate that depends on the technique, one can solve each system of equations for the wage and the price of the relevant capital good. The resulting wage-rate of profits curve for the Alpha technique is:

w_α = (1 - r)/(7 + 5 r)

The maximum wage for the Alpha technique, 1/7 bushels per person-year arises for a rate of profits of zero in the above equation. The maximum rate of profits, for the Alpha technique, is 100% and occurs when the wage is zero.

The wage-rate of profits curve for the Beta technique is:

w_β = (43 - 48 r)/361

For what it is worth, the maximum wage for the Beta technique is 43/361 bushels per person-year. The maximum rate of profits is 43/48, approximately 90%. Both the maximum wage and the maximum rate of profits for the Beta technique are dominated by the corresponding values for the Alpha technique.

Figure 1, at the top of this post, graphs the wage-rate of profits curves for both techniques. Since the coefficients of production in copper-production are a constant multiple (48/91) of the coefficients of production in the process for producing corn from copper, the wage-rate of profits curve for the Beta technique is a straight line. The wage-rate of profits curve for the cost-minimizing technique forms the outer envelope in Figure 1. The Alpha technique minimizes costs for all feasible rates of profits and wages.

One switch point arises in this example. It is at 50%, half the maximum rate of profits for the Alpha technique. The wage is 1/19 bushels per person-year at the switch point, and the slope of both wage-rate of profits curves has a value -48/361 at the switch point. One can find the rate of profits for the switch point by equating the functions for w_α and w_β. A quadratic equation arises for the rate of profits, and 50% is a repeated root for this polynomial. Both the Alpha and Beta techniques are cost-minimizing at the switch point.

4.0 The Market for "Capital"

One can find gross outputs of each process needed to produce a bushel of corn. If the Alpha technique is used, gross outputs consist of two tons iron and 1 bushel corn. For the Beta technique, gross outputs consist of 91/43 tons copper and one bushel corn. The quantity of the capital good, in physical units, needed to produce a net output of one unit of the numeraire good is immediately obvious in this technology. The total quantity of labor, over all processes in a technique, for producing a net output of corn is vector dot product of the labor coefficients, for the technique, and the gross outputs.

The quantity of the capital good must be evaluated with prices so as to graph, say, the amount of capital per person-year for each technique in one space. Since the wage-rate of profits curve for the Alpha technique has some non-zero convexity, the price of iron varies with the given rate of profits:

p_iron = 2/(7 + 5 r)

The price of copper is a constant 48/91 bushels per ton.

Table 2 brings these calculations together. It shows the ratio of the value of the capital good to labor inputs. The horizontal line shows the real Wicksell effect at the switch point. If one wanted, one could remove the price Wicksell effects with Champernowne's chain index measure of capital.

Figure 2: Capital per Worker versus Rate of Profits

5.0 The Labor Market

For completeness, Figure 3 graphs the wage against the amount of labor hired, across all industries, to produce a net output of corn with cost-minimizing techniques. A linear combination of the techniques at the switch point is shown here, also, by a horizontal line.

Figure 3: Labor per Unit Output versus Wage

6.0 Why This Example is a Fluke

Generic results show a certain structural stability. Qualitative properties, for generic results, continue to persist for some small variation in model parameters. This is not the case for the example. Small variations will lead to either two switch points (that is, reswitching) or no switch points. In the latter case, the Beta technique would be dominated and never cost-minimizing.

I look to the mathematics of dynamical systems for an analogy. One can look at prices of production as fixed points in some dynamical system. For example, consider a classical view of competition, in which firms and investors are able to shift from the production in one industry to production in another. (Literature on such dynamical processes can be found under the keyword of "cross-dual dynamics".) Neoclassical economists might look at prices of production as a special case of an intertemporal equilibrium, in which initial endowments just happen to be such that relative spot prices do not vary with time. Or one can consider prices of production as partially characterizing a fixed point, in a limiting process, as time grows without bound in neoclassical models of intertemporal or temporary equilibria.

At any rate, hyperbolic points are considered generic in dynamical systems. In discrete time, no eigenvalues of the linearization around a hyperbolic point lie on the unit circle. Continuing in the jargon, no center manifold exists for a hyperbolic point. Non-hyperbolic fixed points are important in that they indicate a bifurcation.

Example Of The Choice Of Technique

Figure 1: Aggregate Production Function

1.0 Introduction

This long post presents an analysis of the choice of technique in a three-commodity example. This example extends a previous post. Two processes are known for producing each commodity. The example is simple in that it is of a model only of circulating capital. No fixed capital - that is, machines that last more than one period - exists in the model. Homogeneous labor is the only non-produced input used in production.

Despite these simplifications, many readers may prefer that I revert to examples with fewer commodities and processes. Eight techniques arise for analysis. All three commodities are basic in all techniques. I end up with 34 switch points. Even so, various possibilities in the theory are not illustrated by the example. (Heinz Kurz and Neri Salvadori probably have better examples. I also like J. E. Woods for exploring possibilities in linear models of production.) The example does suggest, however, that the exposed errors taught, around the world, to students of microeconomics and macroeconomics cannot be justified by the use of continuously differentiable, microeconomic production functions.

2.0 Technology

This economy produces a single consumption good, called corn. Corn is also a capital good, that is, a produced commodity used in the production of other commodities. In fact, iron, steel, and corn are capital goods in this example. So three industries exist. One produces iron, another produces steel, and the last produces corn. Two processes exist in each industry for producing the output of that industry. Each process exhibits Constant Returns to Scale (CRS) and is characterized by coefficients of production. Coefficients of production (Table 1) specify the physical quantities of inputs required to produce a unit output in the specified industry. All processes require a year to complete, and the inputs of iron, steel, and corn are all consumed over the year in providing their services so as to yield output at the end of the year.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	1/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

A technique consists of a process in each industry. Table 2 specifies the eight techniques that can be formed from the processes specified by the technology. If you work through this example, you will find that to produce a net output of one bushel corn, inputs of iron, steel, and corn all need to be produced to reproduce the capital goods used up in producing that bushel.

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

3.0 Choice of Technique

Managers of firms choose processes in their industry to minimize costs. So one must consider prices in analyzing the choice of technique. Assume that corn is the numeraire. In other words, the price of a bushel corn is one monetary unit. I assume that labor is advanced, and that wages are paid out of the surplus at the end of the year.

These conditions specify a system of three equations that must be satisfied if a technique is to be chosen. For example, suppose the Alpha technique is in use. Let w_α be the wage and r_α the rate of profits. Let p₁ be the price of iron and p₂ the price of steel. If managers are willing to continue producing iron, steel, and corn with the Alpha technique, the following three equations apply:

((1/6)p₁ + (1/200)p₂ + (1/300))(1 + r_α) + (1/3)w_α = p₁

((1/200)p₁ + (1/4)p₂ + (1/300))(1 + r_α) + (1/2)w_α = p₂

(p₁)(1 + r_α) + w_α = 1

These equations apply to iron, steel, and corn production, respectively. They show the same rate of profits being earned in each industry. Confining one's attention to the three processes comprising the Alpha technique, they show the same rate of profits being earned in each industry. Managers will not want to disinvest in one industry and invest in another, at least, with these three processes available.

Suppose the wage is given, is non-negative, and does not exceed a certain maximum specified, for a technique, by a zero rate of profits. Then, for each technique, one can find the rate of profits and prices of commodities. The function relating the rate of profits to the wage for a technique is known as the wage-rate of profits curve, or, more shortly, the wage curve for the technique. Figure 2 graphs the wage-rate of profits curves for the eight techniques in the example.

Figure 2: Wage-Rate of Profits Curves

The cost minimizing technique, at a given wage, maximizes the rate of profits. That is, wage curves for cost minimizing techniques form the outer envelope of the wage curves graphed in Figure 2. Table 3 lists the cost minimizing techniques for the example, from a wage of zero to the maximum wage. The switch points on the frontier are pointed out in Figure 2. This is not an example of reswitching or of the recurrence of techniques. No technique is repeated in Table 3. It is an example of process recurrence. The corn-producing process labeled "e", repeats in Table 3. I label the switch point between Alpha and Beta as "perverse" just to emphasize that results arise for it that violate the beliefs of outdated and erroneous neoclassical economists. From the standpoint of current theory, it is not any more surprising than non-perverse switch points.

Table 3: Techniques on Frontier

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Delta	a, d, f
Theta	b, d, f
Eta	b, d, e

I experimented, somewhat, with the coefficients of production for alternative processes in the various industries, but not all that much. Thirty four switch points exist in the example, including switch points (some "perverse") inside the frontier. No techniques have three switch points, even though in a model with three basic commodities, such can happen. As noted above, no reswitching occurs on the frontier. But consider switch points for each pair of techniques, including within the frontier. Under this way of looking at the example, reswitching arises for the following pair of techniques:

• Alpha and Beta: Vary in corn-producing process.
• Alpha and Delta: Vary in steel-producing process.
• Alpha and Zeta: Vary in iron-producing and corn producing processes.
• Alpha and Theta: No processes in common.
• Gamma and Delta: Vary in corn-producing processes.
• Gamma and Zeta: No processes in common.
• Gamma and Theta: Vary in iron-producing and corn producing processes.

Generically, in models with all commodities basic, techniques that switch on the frontier differ in one process. So one could form a reswitching example with two technique out of the processes comprising, for example, the Alpha and Delta techniques.

Figure 2 is complicated, and some properties of the wage curves are hard to see, no matter how close you look. All wage curves slope downward, as must be the case. The wage curve for, for example, the Alpha technique varies in convexity, depending at what wage you find its second derivative. For high wages, the wage curve for Alpha lies just below Gamma's, the wage curve for Beta is just below Zeta's, and the wage curve for Delta is just below Theta's. (By "high wages", I mean wages larger than the wage for the single switch point for the given pair of techniques.) The wage curves for Epsilon and Eta are visually indistinguishable in the figure. They have a single switch point at a fairly low wage, and above that, the wage curve for Epsilon lies below Eta's. I wonder how much variations in the parameters specifying the technology result in variation in the location of wage curves.

4.0 The Capital "Market" and Aggregate Production Function

The example illustrates certain results that I find of interest. Suppose the economy produces a net output of corn. Given the wage, one can identify the cost minimizing technique. By use of the Leontief inverse for that technique, one can calculate the level of outputs in the iron, steel, and corn industries needed to replace the capital goods used up in producing a given net output of corn. In a standard notation, used in previous posts:

q = (I - A)^-1 (c e₃)

where I is the identity matrix, e₃ is the third column of the identity matrix, c is the quantity of corn produced for the net output, A is the Leontief matrix for the cost minimizing technique at the given wage, and q is the column vector of gross outputs of iron, steel, and corn. (This relationship can be extended to a steady state, positive rate of growth, up to a maximum rate of growth.)

For the given wage, one can find prices that are consistent with the adoption of the cost minimizing technique. Let p be the three-element row vector for these prices. (Since corn is the numeraire, p₃ is unity.) Consider the production of a net output of corn. The column vector of capital goods needed to produce this net output is (A q). The value of these capital goods is:

K = p A q

Let a₀ be the row vector of labor coefficients for the cost minimizing technique.

L = a₀ q = a₀ (I - A)^-1 (c e₃)

Net output per worker is easily found:

y = c/L

Likewise, capital per worker is:

k = K/L

This algebra allows one to draw certain graphs for the example. Figure 3 shows the value of the capital goods the managers of firms want to employ per worker as a function of the rate of profits. As is typical in the Marshallian tradition for graphing supposedly downward-sloping demand functions, the "quantity" variable - that is, the value of the capital goods - is on the abscissa. The "price" variable - that is, the rate of profits - is graphed on the ordinate.

Figure 3: Value of Capital Hired at Different Rates of Profit

Switch points appear in Figure 3 as horizontal lines. They result from varying linear combinations of techniques, at a given price system. The curves, that are not quite vertical, between the switch points result from variations in prices and the rate of profits, for a given cost minimizing technique, with the wage along the wage-rate of profits frontier. Variation from the vertical for these curves is known as a price Wicksell effect.

While it is not obvious from the figure, the sign of the slope of the curve above the switch point between the Alpha and Beta technique changes over the range in which Alpha is the cost-minimizing technique. (This change in the direction of the price Wicksell effect is equivalent to a change in the convexity of the wage curve for the Alpha technique in the region where it lies on the outer frontier in Figure 2.) In the lower part of this uppermost locus, a lower rate of profits is associated with a greater value of capital goods per worker. This is a negative price Wicksell effect. Elsewhere, in the graph, price Wicksell effects are positive. It is not clear to me that neoclassical economists, at least after the Cambridge Capital Controversy, have any definite beliefs about the direction of price Wicksell effects.

The direction of real Wicksell effects cannot be reconciled with traditional neoclassical theory. Consider, first, the switch point between the Theta and Eta techniques. Compare the value of capital per worker at a rate of profits slightly higher than the rate of profits at the switch point with capital per worker at a rate of profits slightly lower. Notice that with this notional variation, a higher value of capital per worker is associated with a lower rate of profits. This is a negative real Wicksell effect. If capital were a factor of production, a lower equilibrium rate of profits would indicate it is less scarce, and firms would be induced to adopt a more capital-intensive technique of production. Thus, a negative real Wicksell effect illustrates traditional, mistaken neoclassical theory. But, in the example, the real Wicksell effect is positive at the switch point between the Alpha and Beta techniques.

I have above outlined how to calculated the value of output per worker and capital per worker as the wage or the rate of profits parametrically varies. Figure 1, at the head of the post, graphs the value of output per worker versus capital per worker. The scribble at the top is the production function, as in, for example, Solow's growth model.

Before considering the details of this function in the example, note that the production function is not a technological relationship, showing the quantity of a physical output that can be produced from physical inputs. Prices must be determined before it can be drawn. In particular, either the wage or the rate of profits must be given to determine a particular point on the production function. Suppose all real Wicksell effects happen to be negative, and the slope of the production function, for some index of capital intensity, happens to be equal to the rate of profits (at each switch point). Since one had to start with the wage or the rate of profits, even then one could not use the production function to determine distribution. Deriving such a marginal productivity relationship seems to be besides the point when it comes to defending neoclassical theory.

Now to details. Between switch points, a single technique lies on the frontier in Figure 2. Given the technique and net output, a certain constant output per worker results, no matter what the wage and the rate of profits in the region where that wage curve lies on the frontier. Thus, the horizontal lines in the graph of the production function reflect a region in which a switch of techniques does not occur. The downward-sloping and upward-sloping lines, in the production function, illustrate switch points. At each switch point, a linear combination of techniques minimizes costs. The perverse switch point is reflected in the production function by an upward slope at the switch point, as the rate of profits parametrically increases. I gather it is a theorem that greater capital per worker is associated with more output per worker. But in the "perverse" case, greater capital per worker is associated with a greater rate of profits.

5.0 The Labor "Market"

So much for neoclassical macroeconomics. Next, consider how much labor, firms want to hire over all three industries, to produce a given net output of corn (Figure 4). (I still follow the Marshallian tradition of putting the price variable on the Y-axis and the quantity variable on the X-axis.) Around the switch point between the Alpha and Beta technique, a slightly higher wage is associated with firms wanting to employ more labor, given net output. In the traditional neoclassical theory, a higher wage would indicate to firms that labor is scarcer, and firms would be induced to adopt less labor-intensive techniques of production. The example shows that this theory is logically invalid.

Figure 4: Labor Employed at Different Wages

6.0 Labor Employed Directly in Corn Production

Although this is not an example of reswitching, it is an example of process recurrence. (I was pleased to see that each of the six production processes is part of at least one technique with a wage curve on the frontier.) Since two processes are available for producing corn, the amount of labor that corn-producing firms want to produce, at a non-switch point, is either 1.0 or 1.5 person-years per gross output of the corn industry. These are the labor coefficients, for processes "e" and "f", in Table 1. The labor coefficients account for the locations of the vertical lines in Figure 5. Once again, a linear combination of techniques is possible at switch points. If the pair of techniques that are cost minimizing at a switch point differ in the corn-producing process, a horizontal line is shown in Table 5. The analysis of the choice of technique is needed to locate these horizontal lines.

Figure 5: Labor Directly Employed in Producing Corn

Figure 5 shows, that around the switch point for the Alpha and Beta techniques, a higher wage is associated with corn-producing firms wanting to hire more labor for direct employment in producing corn. So much for microeconomics. Those exploring the theory of production have found other results that contradict neoclassical microeconomics.

7.0 Conclusion

The above has presented an example in which, in each industry, firms have some capability to trade off inputs, in some sense. For producing a unit output of iron or steel, they might be able to lower labor inputs at the expense of needing to hire more commodities used directly in producing that output. As I understand it, if possibilities of substitution are increased without end, traditional mistaken parables, preached by mainstream economists, are not restored. Suppose the cost-minimizing technique varied continuously along the wage-rate of profits frontier. A specific coefficient of production, as a process varied in some industry, would not necessarily vary continuously. The stories of marginal adjustments that many mainstream economists have been telling for over a century seem to be contradicted by the theory of production.

I have highlighted three results, at least, for the example:

• Around a so-called perverse switch point, a lower rate of profits is associated with firms wanting to adopt a technique in which the value of capital goods, per worker, is less than at a higher rate of profits.
• In the labor market for the economy as a whole, a higher wage can be associated with firms wanting to employ more workers to produce a given (net) output.
• In a given industry, a higher wage can be associated with firms in that industry wanting to employ more workers to produce a given (gross) output.

The last result, at least, is independent of the first. For instance, examples exist of non-perverse switch points in which this result arises.

The theory of supply and demand has been lying in tatters, destroyed for about half a century. Many economists seem to be ignorant of this, though.

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.

Perturbation Of A Reswitching Example

Figure 1: Wage-Rate of Profits Curve for Two Techniques

1.0 Introduction

In this post, I consider a perturbation of the data on technology in this example of the production of commodities by means of commodities. This example is of the choice of technique from two techniques. Each technique can be used to produced a commodity, corn, used for consumption and as the numeraire. The perturbations considered here drastically changes the qualitative characterization of the technology. And they only slightly change the location of switch points and the maximum wages, for the two techniques. These perturbations also only slightly change the maximum rate of profits for one technique. They do, however, drastically lower the maximum rate of profits for the other technique.

2.0 Two Techniques With Two Perturbations

Table 1 displays the technology available to the firms in this example. (I have renamed the industries and commodities.) Each column defines the coefficients of production for a process for producing the output of an industry. Only one process is known for producing iron, and only one process is available for producing steel. Two processes are known for producing corn. Coefficients of production show how much of each input must be available, to provide flows of services of that input over the year, per unit output produced and available at the end of the year. The parameters δ and ε must both be nonnegative for a given technology.

Table 1: The Technology

Inputs	Iron Industry	Steel Industry	Corn Industry
			Alpha	Beta
Labor (Person-Years):	1/3	1/2	1	3/2
Iron (Tons):	1/6	ε	1	0
Steel (Tons):	ε	1/4	0	1/4
Corn (Bushels):	δ	δ	0	0
Output (Various):	1	1	1	1

Two techniques are defined, in this technology, for producing a net output of corn. Each technique consists of a single process for producing corn and whichever of the iron-producing and steel-producing processes (sometimes both) is needed to reproduce the capital goods used up in producing a net output of corn.

2.1 No Basic Commodities

Consider the special case where:

δ = ε = 0

In this case, one can say that in both techniques, no commodity is basic. Or one might say that, in each technique, one commodity is basic, and that which commodity is basic varies with the technique. It depends on how you look at it.

In the Alpha technique, corn is produced with the process labeled Alpha. Iron is used as an input in producing iron and in producing corn. Corn is not an input in any process, and steel is not produced. If one disregarded the non-produced commodity, steel, one could say iron is the single basic commodity. On the other hand, if one included steel as a possible commodity, iron would not be basic, since it does not enter into the production of steel, either directly or indirectly.

The same paragraph could be written about the Beta technique, with the role of iron and steel reversed.

2.2 Three Basic Commodities

Cosider a case in which both the δ and ε parameters are (small) positive numbers. I worked out the following case:

δ = 1/300

ε = 1/200

In this case, the Alpha technique consists of the iron-producing process, the steel-producing process, and the corn-producing process labeled Alpha. All three commodities are basic. Corn enters indirectly into the production of corn through both iron and steel. Similarly, all three commodities are basic in the Beta technique.

So one sees that the structure of production, in both techniques, is qualitatively different in these two cases. This difference is seen in which commodities are basic, and which are not.

3.0 Wage-Rate of Profits Curves

The managers of firms choose the processes comprising the technique so as to minimize cost. Let a bushel of corn be the numeraire. Suppose labor is advanced, and wages are paid out of the output available at the end of the year.

For each technique, these assumptions are such that a relation between the wage and the rate of of profits arises. Both the wage and the rate of profits range between zero and a finite maximum wage or rate of profits. The higher the rate of profits, the lower the wage and vice versa. You can see these wage-rate of profits curves graphed in the first figure here for the special case in which δ = ε = 0. Figure 1, at the top of this post, graphs these wage curves for the specific positive values of δ and ε graphed above.

The choice of technique can be analyzed based on the outer frontier of the wage-rate of profits curves for the technique. For a given rate of profits, the cost-minimizing technique is the one with the highest wage at that rate of profits. At switch points, more than one technique is cost-minimizing. Firms can adopt a linear combination of the techniques on the outer frontier at switch points.

This is a reswitching example for the perturbations considered here. The Alpha technique is cost-minimizing at low and high rates of profits, with the Beta technique cost-minimizing at intermediate rates of profits. Tables 1 and 2 specify the location of the switch points, as well as the maximum wages and rates of profits for the two techniques. These solution values can be found as easy-to-calculate rational numbers for the original case, as shown in Table 1. Table 2 lists approximate values.

Table 1: The Model with δ = ε = 0

Variable	Alpha Technique	Beta Technique
Maximum Wage	5/7 = 0.7143	3/5 = 0.6
Maximum Rate of Profits	500%	300%
First Switch Point
Wage	1/2 = 0.5
Rate of Profits	100%
Second Switch Point
Wage	1/3 = 0.3333
Rate of Profits	200%

Table 1: Perturbed Model

Variable	Alpha Technique	Beta Technique
Maximum Wage	0.7094	0.5991
Maximum Rate of Profits	298.0%	294.1%
First Switch Point
Wage	0.5153
Rate of Profits	84.15%
Second Switch Point
Wage	0.2376
Rate of Profits	231.2%

Small variations in the data defining the technology results in small variations in, for example, the maximum wages and the location of switch points. Decreased requirements for commodity inputs in production processes results in an outward movement of the wage-rate of profits curves and the outer frontier. But some changes resulting from these perturbations of the data are discontinuous. The maximum rate of profits is the most noticeable in this example. When iron is the only input in the iron-producing process, the maximum rate of profits for Alpha is 500%. (This maximum depends only on how much iron is required to produce a unit output of iron.) A perturbation that results in all three commodities being basic in both techniques abruptly lowers this maximum rate of profits to below 300%, the maximum rate of profits in the Beta technique in the original example. I also like that the perturbed model, with three basic commodities, removes the necessity for the convexity of a wage curve to be fixed in one direction for the entire curve.

4.0 Conclusion

This example has illustrated the transformation of a simple reswitching example, through perturbations, to another example, in which all commodities are basic. In this three-commodity example, with all commodities basic, the wage-rate of profits curve for the Alpha technique varies in convexity along its extent. Such a variation in convexity is a general property of multicommodity models of the production of commodities by means of commodity, but cannot be seen in two-commodity examples.

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.)

Teaching Calculus To Kids These Days?

1.0 Introduction

A couple of years ago I saw somebody in my local library who was obviously tutoring students in mathematics. I cannot recall how or why, but I started a question. He assured me that advanced high school seniors were taught calculus here. But the approach they teach nowadays does not require kids to learn epsilon-delta definitions of limits and continuity. This surprised me. I understand limits are difficult to wrap one's mind around. For one thing, one needs to not think in terms of dynamics, in some sense. And epsilonic definitions are rarely seen as natural to the beginning student.

I have since had similar conversations with a few youngsters. And they did not recall epsilon-delta definitions either. I realize that teaching and student recollection varies. Furthermore, the use of epsilon to represent a small distance in the space of the range of the function is a notational convention. Perhaps, some other symbol was used in their classes (although I doubt it). Furthermore, to engineers and practical-oriented students, they might be more interested in getting to problems with derivatives and integrals. (When I asked C. how his calculus class was, he said, "We're still on limits", which I thought expressed an impatience.)

I wonder about this. I have a theory how some might have justified a change to teaching in calculus since my day, although I can imagine other justifications that do not contradict my ideas below. Anyways, I only intend to raise questions in this post.

2.0 A Potted History of Calculus after Newton

When Newton and Liebniz invented the differential calculus, they had a problem with certain quotients. The slope of secants, drawn for two points on a "smooth" function, might be a well-defined ratio. But what does it mean to take a limit? Sometimes Newton seems to treat a denominator as simultaneously zero and non-zero. And this problem with infinitesimals (or fluxions) is compounded when one starts thinking about second derivatives and even higher orders.

Berkeley quickly pointed out these difficulties. I gather he was concerned to argue against the deism - to him, atheism - that often seemed to accompany Newtonian physics and cosmology. Why criticize the mote in your neighbor's eye without first casting out the beam in your own? Anyways, mathematicians recognized Berkeley had a point about calculus. But the mathematics worked in practice and seemed to be extraordinary useful for physics.

So mathematicians struggled for centuries, building an immense structure on what they recognized to be an unsound foundation. They also tried to rebuild the foundations. Cauchy, for example, made some improvements. As far as real numbers and limits are concerned, the decisive work came in the second half of the nineteenth century, with Weierstrass' epsilon-delta definitions and Dedekind's construction of the reals out of sets of rational numbers, known as cuts. Whether this was the answer, or whether this just moved the problems deeper down to questions about sets and logic, was not immediately clear. The work of Cantor, Frege, and Russell are of some importance here. The twentieth century saw intensive exploration of such foundational questions. Anyways, nobody seems to have ever found a contradiction in Zermelo-Fraenkel set theory, even if the absence of such contradictions cannot be proven. ZF set theory, with the axiom of choice in many applications, seems to provide a sufficient foundation for the working mathematician.

I guess that that is how the picture stood around, say, 1960. Newton's own approach to calculus was non-rigorous, but epsilon-delta definitions provide all the rigor introductory students of calculus need. Also, Alfred Tarski had invented something called model theory. Along came Abraham Robinson, who used model theory to develop non-standard analysis. Somehow, nonstandard analysis provides a rigorous justification of infinitesimals. (I wouldn't mind understanding the Löwenheim-Skolem theorem either.)

So maybe it does make sense to teach calculus, without the rigor of epsilon-delta definitions. Keisler wrote a textbook to illustrate the teaching of calculus on the foundations of infinitesimals, maybe easier for the student to understand and justified by the rigor of the advanced abstractions of non-standard analysis. Has this approach, revolutionizing centuries of understanding, won out in introductory calculus classes?

3.0 Other Special Cases in Introductory Teaching

I can think of a couple of other cases where what was in my textbooks in calculus and analysis was superseded, in some sense, in more advanced mathematics. I gather mathematical analysis is often informally defined as what the differential and integral calculus would be if taught rigorously. And Rudin (1976) is a standard introduction to analysis.

Rudin provides an epsilon-delta definition of limits. This definition is more general than you might see in (old?) calculus courses. In such less abstract courses, you might see two definition of limits. One would be for sequences, that is, for functions mapping the natural numbers into the reals. And another would be for functions mapping the real numbers into the real numbers. But Rudin's definition is for functions mapping an arbitrary metric space into (possibly another) arbitrary metric space. One might get the impression that some notion of distance between points is needed to define a limit. But, as was pointed out in the class I took with Rudin as the textbook, a limit of a function is a topological notion.

A common intuition for integration is as of the area under a curve. This notion can be formalized with the Riemann integral, and, for me, this is the first definition I learned. But another definition, Lebesque integration, is taught in classes on measure theory. Lebesque integrals are more general. Some functions have a Lebesque integral, but not a Riemann integral. But, if a function has a Riemann integral, it has the same value for the Lebesque integral.

I offer a suggestion in the spirit of a devil's advocate. Why teach the special case at all in these instances? Why not start with the more general case? Do those who concern themselves with the pedagogy of mathematics selectively advocate the teaching of the more abstract, general case? Is so, how do they choose when this is appropriate?

4.0 Conclusion

Is it now quite common - maybe, in the United States - to teach introductory calculus without providing an epsilon-delta definition of a limit? If so, does common justification of this practice draw on a non-standard analysis approach to calculus? Why should this extremely abstract idea influence introductory teaching, but not other abstractions?

Appendix: Two Definitions of a Limit of a Function and a Theorem

These are from memory, since I do not want to bother looking them up. The proof of the theorem, probably stated more rigorously, was a test question in a course I took decades ago.

Definition (Metric Space): Let f be a function mapping a metric space X into a metric space Y. L is a limit of f as x approaches x₀ if and only if, for all ε > 0, there exists a δ > 0 such that, whenever the distance between x and x₀ is less than δ, the distance between f(x) and L is less than ε.

Definition (Topological): Let f be a function mapping a topological space X into a topological space Y. L is a limit of f as x approaches x₀ if and only if for all open sets B in Y containing L, the preimage of B, f^-1(B), contains x₀.

Theorem: Let f map a metric space X into a metric space Y. Then L is a limit of f as x approaches x₀, in the metric space definition, if and only if L is also the limit of f, in the topological space definition, in the topologies for X and Y induced by the respective metrics for these spaces.

References

• George Berkeley. (1734). The Analyst: A Discourse Address to an Infidel Mathematician... [I never finished this.]
• H. Jerome Keisler (1976). Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt. [I barely started this.]
• Morris Kline (1980). Mathematics: The Loss of Certainty, Oxford University Press.
• Walter Rudin (1976). Principles of Mathematical Analysis, 3rd edition, McGraw-Hill.