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.

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.

Why Republicans in the USA are "The stupid party"

1.0 Introduction

In 1865, John Stuart Mill, when he was almost 60, was elected to Parliament. He represented the radical wing of the Liberal party. He had been a public intellectual for decades, with lots of books, editorials, and articles for the Tories to draw on in attacking him. Some Tories overreached. This led to the conservative party becoming known as "The stupid party".

2.0 Adventures in Parliament

I find Mill's attitude towards being a Member of Parliament (MP) unusual, albeit consistent with his stated opinions. He was not interested in giving speeches in support of his party's view when many others were willing to do so. He "in general reserved [him]self for work which no others were likely to do." (from his Autobiography. Uncited quotes below are from this book.) He had such opportunities, for few radicals were in Parliament. (Earlier in his life, such a group was known in Britain as the Philosophical Radicals.)

Despite his radicalism, some of his advocacy was in opposition "to what then was, and probably still is, regarded as the advanced liberal opinion". For example, Mill was against abolishing capital punishment and "in favour of seizing enemies' goods in neutral vessels".

But other efforts seem more progressive, when viewed from the standpoint of later times. In a speech on Gladstone's Reform Bill, Mill argued for sufferage of the working class. He also promoted women's sufferage through his parliamentary work. He put out a pamphlet for reforming British rule in Ireland, including "for settling the land question by giving to existing tenants a permanent tenure, at a fixed rent." He joined in an organization that attempted to have British officers in Jamaica prosecuted, in a criminal case. These officers had engaged in killing, flogging, and general brutality, under the pretence of having civilians brought before court-martials.

3.0 Considerations on Representative Government

J. S. Mill had long been what we would call a public intellectual. I want to particularly focus on his book with the above title. He gives a qualitative discussion of particular voting games. Mill was for proportional representation, also known then as "personal representation". And Mill recommended Thomas Hare on the topic. Other issues he considered include:

• Provide multiple votes (a greater weight) to more highly educated members of the electorate.
• Giving voters multiple votes for distributing in elections for a district that had multiple members to elect to a council.
• Working class and women's sufferage.
• The advantages and disadvantages of a secret ballot (as opposed to an open one).
• The advantages and disadvantages of having a two-stage election (e.g., the electoral college, Senators being elected by a state's legislature.
• The advantages and disadvantages of an upper house (e.g., the Senate, the House of Lords), under various assumptions about its composition.
• Whether or not the chief executive should be independently elected (e.g., the President of the United States) or by the legislature (e.g., the Prime Minister in the United Kingdom).
• How the central government and localities should interact and what should the authority and responsibility of each be.

In short, Mill seems to write about concerns often of interest today in analytical political science, albeit in a qualitative way and grounded in concrete practices in his time.

4.0 Attention and the Aftermath

The Tories in Parliament took advantage of Mill's long paper trail. In debates, they would ask if he wanted to defend some of his previous written statements. Because of Mill's forthrightness, this strategy backfired:

"My position in the House was further improved... by an ironical reply to some Tory leaders who had quoted against me certain passages of my writings, and called me to account for others, especially for one in 'Considerations on Representative Government,' which said that the Conservative party was, by the law of its composition, the stupidest party. They gained nothing by drawing attention to the passage, which up to that time had not excited any notice, but the sobriquet of 'the stupid party' stuck to them for a considerable time afterwards."

Considerations on Representative Government contains this passage:

"...It is an essential part of democracy that minorities should be adequately represented. No real democracy, nothing but a false show of democracy, is possible without it.

Those who have seen and felt, in some degree, the force of these considerations, have proposed various expedients by which the evil may be, in greater or lesser degree, mitigated. Lord John Russell, in one of his Reform Bills, introduced a provision that certain constituencies should return three members, and that in these each elector should be allowed to vote only for two; and Mr. Disraeli, in the recent debates, revived the memory of the fact by reproaching him for it, being of opinion, apparently, that it befits a Conservative statesman to regard only means, and to disown scornfully all fellow-feeling with any one who is betrayed, even once, into thinking of ends."

And that passage has this footnote (which I read as noting the existence of negative partisanship):

"his blunder of Mr. Disraeli (from which, greatly to his credit, Sir John Pakington took an opportunity soon after of separating himself) is a speaking instance, among many, how little the Conservative leaders understand Conservative principles. Without presuming to require from political parties such an amount of virtue and discernment as they that they should comprehend, and know when to apply, the principles of their opponents, we may yet say that it would be a great improvement if each party understood and acted upon its own. Well would it be for England if Conservatives voted consistently for every thing conservative, and Liberals for every thing liberal. We should not then have to wait long for things which, like the present and many other great measures, are eminently both the one and the other. The Conservatives, as being by the law of their existence the stupidest part, have much the greatest sins of this description to answer for; and it is a melancholy truth, that if any measure were proposed on any subject truly, largely, and far-sightedly conservative, even if Liberals were willing to vote for it, the great bulk of the Conservative party would rush blindly in and present it from being carried." (emphasis added.)

I assume Mill's refers to the following statement, in parliamentary debates, as his "ironical reply":

"I did not mean that Conservatives are generally stupid; I meant, that stupid persons are generally Conservative. I believe that to be so obvious and undeniable a fact that I hardly think any honourable Gentleman will question it."

5.0 Conclusion

And so, to this day, the more conservative party in some countries, such as the United States, is sometimes called "The stupid party".

References

• J. S. Mill (1861). Considerations on Representative Government
• J. S. Mill (1873). Autobiography of John Stuart Mill