Variation in Standard Commodity with Relative Markups

I am not sure about the economic logic in this post. Maybe somebody like D'Agata or Zambelli could do something with this. These ideas were suggested to me by email with a sometime commentator.

I start out with notation for Sraffa's price system, modified in an unusual way to allow for persistent variations in the rate of profits among industries:

• a₀ is a row vector of labor coefficients in each of n industries.
• A is a Leontief input-output matrix, where a_{i, j} is the quantity of the ith commodity needed as input to produce one unit of the jth commodity.
• S is a diagonal matrix, where all off-diagonal elements are zero. s_{j, j} is the markup on non-labor costs in the jth industry.
• p is a row vector of prices.
• w is the wage.
• r is the scale factor for the rate of profits.

The coefficients of production, as expressed in the labor coefficients and the Leontief matrix are given parameters. Relative markups are also taken as given. Prices, the wage, and the scale factor for the rate of profits are the unknowns to be determined. My problem is to find a numeraire such that the wage and the scale factor for the rate of profits trade off in a straight-line relationship, at least when labor is advanced and wages are paid out of the net product:

r = R (1 - w)

I assume all elements of A are non-negative and that all elements of a₀ and all diagonal elements of S are positive. The economy is assumed to be viable, that is, as capable of producing a surplus product. For simplicity, assume that the Leontief matrix is indecomposable. More generally, I need A S to be a Sraffa matrix.

For my purposes here, I formulate price equations as so:

p A S (1 + r) + a₀ w = p

Consider the case when wages are zero and the scale factor for the rate of profits is at its maximum R:

p A S (1 + R) = p

Or:

p A S = (1/(1 + R)) p

I observe that prices are a left-hand eigenvector of the matrix A S, with (1/(1 + R)) the corresponding eigenvalue. To ensure that prices are positive, of the n eigenvalues, choose the maximum. The maximum eigenvalue is also known as the Perron-Frobenius root of A S.

Let y^* be a right-hand eigenvector of A S corresponding to its Perron-Frobenius root. Let q^* be gross output such that the net output is y^*:

y^* = q^* - A q^*

These quantities flow define the standard system here, when scaled so as employ a unit quantity of labor:

a₀ q^* = 1

The net output of the standard system is the desired numeraire:

p y^* = 1

With this definition of the standard system, the ratio of physical gross outputs to circulating capital inputs varies among commodities. This result contrasts with Sraffa's standard system. I suppose I could restore this property by choosing q^*, not y^*, to be an eigenvector. Either way, the ratio of net outputs to circulating capital inputs varies among industries. Either way, the relative ratios of commodities in the standard industry depends on relative markups.

Do Marx's invariants hold with the above definition of the standard system? I expect not. Nevertheless, does this mathematics provide some insight into classical or Marxist political economy?

A Fluke Case Over The Wage Axis

Figure 1: Wage Curves and The Price of Corn for the Fluke Case

Introduction

This post extends a previous post. I am basically introducing structural dynamics into an example, by Bidard and Klimovsky of fake switch points.

At a rate of profits of zero in the example, the price of corn is zero for Alpha, one of the two techniques that is cost-minimizing there and for somewhat higher rates of profits. At a time before the fluke case, only the Gamma technique is cost-minimizing at a rate of profits of zero. The price of corn, as calculated with the Alpha technique, is negative at a rate of profits of zero. Alpha prices become non-negative only for positive rates of profits. This possibility cannot arise in examples with only single production and the choice of technique analyzed by the construction of the wage frontier.

2.0 A Fluke Case

Technology and techniques are specified as in the previous post. I consider variations in labor coefficients with time. Two commodities can be produced jointly with each of three production process. In each process, workers produce outputs of the two commodities from smaller inputs of each commodities. Requirements of use are such that at least two processes must be operated. So each technique combines two processes.

A system of price equations is associated with each technique. The system, including an equation specifying the numeraire, can be taken to define the wage and the prices of both commodities, given an exogenous specification of the rates of profits. Table 1, at the head of this post, illustrates the solution prices at a given point of time. Linen is taken as the numeraire. The top half of the figure shows the wage, for each technique, as a function of the rate of profits. The bottom half of diagram shows the corresponding price of corn. Notice that, for the Alpha technique, the price of corn is zero when the rate of profits is zero.

A technique is only feasible, for the analysis of the choice of technique, when both the wage and prices are non-negative. In Figure 1, the rate of profits is partitioned into two roman-numbered regions. In Region II, both the Alpha and Gamma techniques are feasible. In Region III, only the Alpha technique is feasible.

At a switch point:

• The wage curves for at least two techniques intersect at the switch point.
• No extra profits can be made at the going rate of profits in any process.
• No excess costs arise for any process that can be operated at the switch point.

No switch points exist in the example at the time illustrated in Figure 1. For the structure of the example, all three wage curves intersect at a (non-fake) switch point. Furthermore, the price of corn is the same for all three techniques at the switching rate of profits.

Not enough information has been given so far to determine which techniques are cost-minimizing at each feasible rate of profits in Figure 1. I like to plot extra profits for each process and each price system. I do not show such plots in the post. Nevertheless, Table 1 summarizes which techniques are cost minimizing.

Table 1: Cost-Minimizing Techniques

Regions	Cost-Minimizing Technique	Processes
I	Gamma	b, c
II	Alpha and Gamma	a, b, c
III	Alpha	a, c

3.0 Before the Fluke Case

Consider time before the fluke case illustrated in Figure 1. Labor coefficients are larger. Figure 2, below, illustrates the wage and price curves for a specified time before the fluke case described above. Notice the appearance of Region I, where Gamma is uniquely cost-minimizing. The fluke case is a knife-edge case where Region disappears. The wage axis becomes the boundary between Regions I and II. Of these two regions, only Region II exists for a positive rate of profits.

Figure 2: A Fluke Fake Switch Point?

Does Figure 2 illustrate another fluke case? At the fake switch point at a rate of profits of five percent, the price of corn is zero. But consider Figure 3 below. The only difference in the example between Figures 2 and 3 is the specification of the numeraire. With corn as numeraire, the fake switch points disappear, and a new fake switch point appears at a rate of profits of 13 1/3 percent. The wage and the price of linen approach an asymptote for the rate of profits at which the price of corn is zero when linen is the numeraire. Which techniques are cost-minimizing is unaffected by the choice of the numeraire.

Figure 3: Not A Fluke With Corn As Numeraire

4.0 After the Fluke Case

Consider some time after the fluke case illustrated in Figure 1. With the chosen parameters, labor coefficients have decreased less in the first production process than in the other two. Figure 4 shows the next qualitative change in the example, in which a switch point appears over the wage axis. I have already analyzed this case for this example.

Figure 4: A Switch Point On The Wage Axis

Between the times illustrated by Figures 1 and 4, Regions II and III continue to characterize the range of feasible non-negative rates of profits. The price of corn is positive, for all three techniques, is positive for feasible rates of profits for each technique. Region I has vanished.

The switch point continues to exist after the time illustrated in Figure 4, but at a positive rate of profits. A new region appears. For a rate of profits of zero and small positive rates of profits, the Beta technique is uniquely cost-minimizing.

5.0 Conclusion

This post has presented a fluke case only possible under joint production. In this example, the choice of technique cannot be determined by constructing the wage frontier.

This post has also presented a sort-of fluke case associated with a fake switch point. In this case, the fake switch point appears on the frontier at a rate of profits at which the price of corn is zero. The set of cost-minimizing techniques and processes varies at the fake switch point. But its existence depends on the choice of the numeraire.

I have been working on a taxonomy of fluke switch points for understanding structural economic dynamics. This post illustrates that my approach can extend to joint production. New phenomena and fluke cases can arise, and one must, perhaps, pay closer attention to what is and is not dependent on the choice of the numeraire.

A Pattern Over The Wage Axis In A Case Of Joint Production

Figure 1: Wage Curves with Corn as Numeraire

1.0 Introduction

This post presents an example of a fluke switch point in which the choice of technique cannot be analyzed by the construction of the wage frontier. Under joint production, the technique that is cost-minimizing, for a given rate of profits, does not necessarily maximize the wage. Nevertheless, one can still see what I call a pattern over the wage axis in this case. The example is a generalization of the numerical example in Bidard & Klimovsky (2004).

2.0 Technology

I postulate an economy in which two commodities, corn and linen, can be produced from inputs of corn, linen, and labor. Managers of firms know of three processes (Tables 1 and 2) to produce corn and linen. Each process produces net outputs of corn and linen as a joint product. Inputs and outputs are specified in physical units (say, bushels and square meters) per unit level of operation of the given process. Inputs are acquired at the start of the year, and outputs are available for sale at the end of the year.

Table 1: Inputs for The Technology

Input	Process
	(a)	(b)	(c)
Labor	eσ0,1(1 - t)	eσ0,2(1 - t)	eσ0,3(1 - t)
Corn	20	20	30
Linen	20	20	30

Table 2: Outputs for The Technology

Output	Process
	(a)	(b)	(c)
Corn	21	23	36
Linen	27	25	34

I assume that requirements for use are such that two processes must be operated to satisfy those requirements. I need to investigate the implications of this assumption further. Apparently, for this example, it implies that the economy is not on a golden rule steady state growth path, with the rate of profits equal to the rate of growth. Anyway, with this assumption, three techniques - Alpha, Beta, and Gamma - can be operated. Table 3 specifies which processes are operated for each technique.

Table 3: Techniques

Techniques	Processes
Alpha	a, b
Beta	a, c
Gamma	b, c

The technology, as I have defined it, is parameterized. I consider the following specification for the rate of decrease in labor coefficients.

σ_0,1 = 2

σ_0,2 = σ_0,3 = 5/2

Bidard & Klimovsky's example arises when t is unity. I consider the following value for time:

t ≈ 0.91973

Structural economic dynamics arises as time varies.

3.0 Price System

Prices of production arise for each technique and each specification of the numeraire. For the Alpha technique, prices of production are characterized by the system of the following three equations:

(20 p₁ + 20 p₂)(1 + r) + [ e^{σ_0,1(1 - t)} ] w = 21 p₁ + 27 p₂

(20 p₁ + 20 p₂)(1 + r) + [ e^{σ_0,2(1 - t)} ] w = 23 p₁ + 25 p₂

p₁ d₁ + p₂ d₂ = 1

where:

• r is the rate of profits.
• w is the wage.
• p₁ is the price of corn.
• p₂ is the price of linen.
• d₁ is the quantity of corn in the consumption basket serving as numeraire.
• d₂ is the quantity of linen in the consumption basket serving as numeraire.

Given one of the distributive variables, this system of equations can be solved. Figure 1, at the top of this post, graphs the wage curves for the three techniques, when d₁ = 1 and d₂ = 0. Figure 2 graphs the wage curves when linen is the numeraire.

Figure 2: Wage Curves with Linen as Numeraire

Notice that which technique lies on the outer envelope, as the rate of profit, varies with the choice of numeraire. In Figure 1, the Alpha technique maximizes the wage, for all feasible positive rates of profits. In Figure 2, the Gamma technique, then the Alpha technique, maximizes the wage, with an increasing rate of profits. This dependence of qualitative characteristics of the wage frontier cannot arise when all capital goods are circulating capital.

In the example, the two processes for a technique and the remaining process must all obtain the same rate of profits at a (genuine, non-fake) switch point. In the example, all three wage curves must intersect at a switch point. Another aspect of a switch point is that the prices of each good must be invariant across the price systems for the techniques entering the switch point. When corn is the numeraire, the price of linen must be the same for all three techniques at the switch point. This property is illustrated in Figure 3. The corresponding property for the price of corn, when linen is the numeraire, is illustrated in Figure 4. No sign of the fake switch points appears in Figures 3 and 4.

Figure 3: Price of Linen with Corn as Numeraire

Figure 4: Price of Corn with Linen as Numeraire

4.0 Choice of Technique

Wage curves can be misleading when analyzing the choice of technique under models of joint production. How then should the choice of technique be found?

First, suppose the Alpha technique has been adopted. One can cost up the outputs and inputs of each process, for the solution to the price system for the Alpha technique. Figure 5 shows results. No extra profits, sometimes called pure economic profits, are made in operating the processes comprising the Alpha technique. For positive rates of profits, operating process c will not obtain the going rate of profits. Clearly, the Alpha technique is cost-minimizing for the graphed range of the rate of profits.

Figure 5: Extra Profits with Alpha Prices

Second, suppose the Beta technique is chosen. Figure 6 graphs extra profits, for each process, as a function of the rate of profits, given Beta prices. For a positive rate of profits, the second process earns extra profits and will be adopted by managers of firms. Notice that one cannot tell from the diagram which process will be dropped. This issue does not arise without joint production. In the case of single production, only one process in the given technique produces the same commodity as that produced by the new process.

Figure 6: Extra Profits with Beta Prices

Finally, suppose the Gamma technique is chosen. Figure 7 graphs extra profits for this case. And the Gamma process is cost-minimizing for the full range of the rate of profits shown in the figure.

Figure 7: Extra Profits with Gamma Prices

The above has shown that, in this example, both the Alpha and Gamma techniques are cost-minimizing at feasible positive rates of profits. The Beta technique is cost-minimizing only at the switch point at a rate of profits of zero percent. The choice of technique is independent of the numeraire. Presumably, the choice between the Alpha and Gamma techniques is made based on requirements for use. At any rate, the chosen technique need not maximize the wage, given the rate rate of profits and the specification of the numeraire.

5.0 Conclusion

This example has illustrated that a specific fluke switch point, which I originally defined for cases with only circulating capital, can also arise in joint production. I except to find a need for new kinds of fluke switch points as I further examine joint production. I am hoping to be able to draw pattern diagrams in which qualitative properties are independent of the choice of the numeraire.

References

• Bidard, Christian and Edith Klimovsky (2004). Switches and fake switches in methods of production. Cambridge Journal of Economics. 28 (1): 89-97.

Martin Weitzman's The Share Economy

I happen to have one book by Marty Weitzman (1942 - 2019) on my bookshelf. So I thought I would write a bit about The Share Economy: Conquering Stagflation.

This is an ill-timed book. It proposes that firms negotiate with workers to pay them a percentage of revenues, instead of, say, an hourly money wage. It argues that such a change will address the widespread macroeconomic problem, throughout the 1970s, of simultaneously high unemployment and high inflation. But, by the time the book came out, stagflation had been "solved", in an extremely reactionary way. The countervailing power of organized labor was being abolished. Labor unions were being crushed, and workers would, by and large, no longer see their wages increase with productivity. Instead of unemployment being addressed, workers would just have to get used to long-lasting higher unemployment.

Maybe some day, we will get back to a setting where Weitzman's book is socially relevant. Even so, it is worth exploring how macroeconomic performance is affected by microeconomic structures.

Although I think of Weitzman as a mainstream economist, his view of the microeconomic setting at the time of his writing was not that far away from Post Keynesianism. He thinks of the "tone" of "modern industrial capitalism" as set by "a relatively small number of large-scale firms", such as those in the Fortune 500. These firms are described by the theory of monopolistic competition. (quotes on p. 11). These firms are characterized by constant costs over a wide range of levels of production below limits set by capacity. They set their prices at a markup over cost. The theory of profit maximization, under these assumptions, yields a markup based on elasticity of consumer demand.

Weitzman explicitly rejects a theory of monopsony for labor markets:

"...If your aim is to focus in on fine close-up details and you wish to do justice to the facts, you must rely on a heavily institutional approach. But I think the unique long-run substitutability of labor among different uses actually makes the competitive theory a rather good description of long-run tendencies in the labor market...

In this book I am primarily interested in the general theory of wage determination... ...at least the labor market behaves 'as if' it is competitive, in the sense that countervailing power between buyers and sellers of labor is sufficiently balanced that neither party has a clear upper hand and both possess approximately equal bargaining strength. The economy-wide real wage is not very different from what would be determined by competitive forces in the labor market." (pp. 29-30.)

I am not sure that Weitzman's account of firms is consistent with firms operating multiple plants and producing multiple products. I think of Alfred Eichner's theory of the megacorp here. I also doubt that theories of full cost, markup, or administered prices should be developed based on markups determined by elasticities. Rather, the markup might be theorized as based on firm's plans for growth.

Weitzman sees that firms will respond to fluctuations of demand by adjusting quantities, not prices. He cites Janos Kornai's contrast of planned, socialist economies with capitalist economies. In the United States, firms must attend to making the consumer's shopping experience as pleasant as possible, while in the Soviet Union, establishments do not care and consumers wait in queue. On the other hand, establishments in the Soviet Union cater to the worker. Weitzman argues his share economy would change the dynamics of the labor market such that firms in the United States would also worry more about the worker's experience.

Wietzman sees the contemporary practice of firms awarding year-end bonuses as a start towards his share economy. He includes Eastman Kodak as an example. Kodak is now bankrupt, and Kodak Park in Rochester, NY, is mostly empty and decaying. In my anecdotal experience, bonuses are often experienced as a present that cannot be planned or depended on. Maybe it would be different with more transparency from your employer, as resulting from a union contract, representatives from the union sitting on the board of directors, an Employee Stock Ownership Plan (ESOP), or some such.

Overall, I find The Share Economy intriguing. It illustrates how good economists will not develop an universal theory, but will address problems of the economist's own time and place.

(A propos of nothing in particular, Branko Milanovic has a post coming close to an endorsement of Neo-Ricardianism.)

• Martin L. Weitzman (1983). The Share Economy: Conquering Stagflation. Harvard University Press.

Elsewhere

This list is mostly a matter of aspirational reading.

• Maybe I want to read Ted Burczak's Socialism after Hayek. (The Amazon page has one of Herb Gintis' long reviews.)
• Binyamin Appelbaum's The Economists' Hour: False Prophets, Free Markets, and the Fracture of Society is not even out yet, and already some mainstream economists are whining about it on twitter.
• William R. Clark and Vincent Arel-Bundock have a paper, Independent but not indifferent: Partisan bias in monetary policy at the Fed, in 2012.
• Christopher Gandrud and Cassandra Grafström's Does presidential partisanship affect Fed inflation forecasts? is related.
• Mark Buchanan recommends that more attention be paid to the work of Ole Peters and others at the London Mathematical Laboratory. They have developed something called ergodicity economics, as a replacement for expected utility theory.