Thursday, May 21, 2020

More On A Fixed Capital Example

 Figure 1: A Partition of a Parameter Space for the Schefold Example
1.0 Introduction

I want to revisit a perturbation analysis of an example, from Bertram Schefold, of reswitching with fixed capital.

Suppose workers use a machine to produce something or other, where the machine lasts several production periods. It is a possible choice to run the machine for less than its full physical life. One might think than choosing to adopt a technique with a longer economic life of the machine is, in some sense, more capital-intensive than choosing to junk it sooner. And this will lead to more output per worker. But, I am surprised to say, this is just not so. The underlying vision of Austrian-school economists is wrong even here.

If I worked through all the problems, with full understanding of their implications, in Kurz and Salvadori (1995), I would probably already know this. But, at least, I can make this point by examining perturbations of a single numeric example.

2. Technology

Table 1 shows the coefficients of production for the three processes comprising the available technology. Inputs must be available at the beginning of the year, and outputs become available at the harvest at the end. In the first process, labor uses inputs of corn to produce a new machine. That machine is used by labor in the second process, with inputs of seed corn, to produce more corn and a one-year old machine. In the third process, labor uses inputs of seed corn and the one-year old machine to produce corn. (I did think of calling the machine a "tractor".) The machine varies in physical efficiency over the course of its lifetime. In its second year, it requires less labor to tend it, but more inputs of corn.

 Input Process (I) (II) (III) Labor (1/10) e1 - σ t (43/40) e1 - φ t e1 - φ t Corn (1/16) e1 - σ t (1/16) e1 - φ t (1/4) e1 - φ t New Machines 0 1 0 Old Machines 0 0 1 Outputs Corn 0 1 1 New Machines 1 0 0 Old Machines 0 1 0

I postulate the possibility of technical progress. The parameter σ specifies the rate of technical progress in using machines, while φ specifies the rate of technical progress in using machines to produce corn. When (σ t) and (φ t) are unity, this is a reswitching example from Bertram Schefold (1980: 179). Corn is assumed to be the only consumer good, and it also is used as circulating capital. I also take a unit of corn as the numeraire.

A choice of technique arises. I call Alpha the technique in which the machine is operated only one year, and I assume free disposal. Beta is the technique in which the machine is operated for its full physical life time.

3.0 Price Equations

I take corn as numeraire. Although I do not make use of this generality in this post, I formulate systems of equalities and inequalities for prices of production with the possibility of persistent differences in the rate of profits among processes. p1 is the price of a new machine, p2 is the price of an old machine, w is the wage, and r is the scale factor for the rate of profits. The relative rates of profits are specified by given ratios s1, s2, and s3 for the processes, which I take to be unity throughout. Coefficients of production are taken as given; one might think of the parameter t as expressing a very slow, secular time. With these parameters, and either the wage or the scale factor for the rate of profits taken as given, one solves for the prices of new and old machines, the other distributive variable, and the choice of technique.

3.1 Price Equations for the Alpha Technique

Suppose the Alpha technique is cost-minimizing, and the machine is discarded after being used for one year. The price of a new machine, the wage, and the scale factor for the rate of profits are related by the following two equations:

[(1/16) e1 - σ t](1 + s1 r) + [(1/10) e1 - σ t] w = p1

[(1/16) e1 - φ t + p1](1 + s2 r) + [(43/40) e1 - φ t] w = 1

The first equations comes from the process for producing a new machine. The second equation comes from the process for producing corn with a new machine. The price of an old machine is zero:

p2 = 0

For the Alpha technique to be cost-minimizing, extra profits must not be available from operating the process that produces corn with an old machine. That is, the cost of a unit level operation of that process must not fall below revenues:

[(1/4) e1 - φ t](1 + s3 r) + [e1 - φ t] w ≥ 1

A switch point exists when the above equation is met with equality.

3.2 Price Equations for the Beta Technique

Now suppose the Beta technique is cost-minimizing. The prices of a new and old machine, the wage, and the scale factor for the rate of profits are related by three equations:

[(1/16) e1 - σ t](1 + s1 r) + [(1/10) e1 - σ t] w = p1

[(1/16) e1 - φ t + p1](1 + s2 r) + [(43/40) e1 - φ t] w = 1 + p2

[(1/4) e1 - φ t + p2](1 + s3 r) + [e1 - φ t] w = 1

For the Beta technique to be cost-minimizing, the price of an old machine must not be negative.

p2 ≥ 0

Here, too, a switch point exists when the above equation is met with equality. Suppose prices of production for the Beta technique yield a negative price for an old machine. This is a signal to firms that they should truncate the economic life of the machine. They should only operate it for one year.

4.0 Perturbations

I do my usual thing of analyzing how the choice of technique varies with technical progress along a secular path, given values of parameters that specify the rate of decrease in specified coefficients of production. Technical progress in the example reduces the inputs of labor and circulating capital needed to produce a given output. The notation allows the rate of technical progress to differ between the machine and corn industries. The rate of technical progress is assumed to affect both processes that produce corn in the same way, whether or not the machine the workers use is new or old.

4.1 A Temporal Path

The choice of technique varies with distribution and technical progress. Figure 2 illustrates one particular ratio of the rate of technical progress in producing machines to the rate in using machines. Changes in the number and characteristics of switch points occur with patterns already described in previous blog posts. Switch points appear over the axis for the rate of profits and over the wage axis. Schefold's particular parameterization occurs just before switch points vanish in a reswitching pattern.

 Figure 2: A Case of Variation in Switch Points with Technical Progress

4.2 Another Temporal Path

Figure 3 illustrates another particular ratio of the rate of technical progress in producing machines to the rate in using machines. Here too, the technical progress in using machines overwhelms technical progress in producing machines, and this is manifested by a transition ultimately from a region in which the economic life of a machine is one year to a region in which its economic life is its whole physical life, whatever the distribution of income. In this example of technical progress, a pattern over the wage axis precedes a pattern over the axis for the rate of profits. A reswitching example arises and disappears. It is then cost-minimizing to run the machine for its full physical life at high and low wages, but at intermediate wages cost-minimizing firms will truncate the economic life of the machine.

 Figure 3: Another Case of Variation in Switch Points with Technical Progress

4.3 A Partition of the Parameter Space

At each point on the two paths through logical time characterized by Figures 2 and 3, the amount of technical progress in producing machines and in using machines is specified. Figure 1 partitions the resulting parameter space, based, as usual, on the number and characteristics of switch points in the choice of technique. Table 2 notes the switch points and lists the cost-minimizing technique, from a low to a high wage, for each numbered region. The locii separating Regions 1 and 2 and Regions 3 and 4 are parallel affine functions. The locus for the reswitching pattern is tangent to the locus for the pattern over the wage axis at the intersection of Regions 2, 3, and 4. Figure 1 demonstrates the applicability of the taxonomy of fluke switch points to the case of fixed capital.

 Region Switch Points Cost-Minimizing Techniques 1 None Alpha 2 One Beta, Alpha 3 Two Beta, Alpha, Beta 4 None Beta 5 One Alpha, Beta

The Schefold example is in narrow edge at the left of Region 3, along the dotted line corresponding to Figure 2. Just as this approach of partitioning (projections of) parameter spaces can be used to construct fluke switch points, it can also be used to improve examples of non-fluke switch points. Other parameters in Region 3 can result in switch points further apart on the wage frontier. The price of an old machine, at prices of production for the Beta technique, can be more noticeably negative when the Alpha technique is cost-minimizing.

5.0 Compare and Contrast Regions 2 and 5

Comparing and contrasting the single switch point in Regions 2 and 5 is interesting. Figures 4 and 5 illustrate the application of the direct method to analyze the choice of technique. When the machine is junked after one year, the price of an old machine is zero. Figure 4 shows extra profits in operating the machine for the second year with prices of production in this case. Around the switch point in Region 2, it pays to run the machine for a second year for higher rates of profits. On the other hand, around the switch point in Region 5, it pays to run the machine for the second year at lower rates of profits. Figure 5 shows the price of the old machine, given prices of production, when the machine is run for two years. A negative price shows that the cost-minimizing firm should truncate the economic life of machine to one year. Both figures yield the same conclusion about the choice of technique in the two regions. (This consistency is guaranteed in a pure fixed capital example with no superimposed joint production.)

 Figure 4: Regions 2 and 5 with Price of Old Machine of Zero

 Figure 5: Price of Production when Machine Runs for Two Years

Around the switch points in both Regions 2 and 5, a lower wage is associated with cost-minimizing firms wanting to hire more labor, given net output (Table 3). As one might expect, the adoption of a more labor-intensive technique is associated with lower net output per worker. It seems that the economic life of a machine cannot be mapped to the capital-intensity of a technique. Adopting a technique in which a machine is run longer is not necessarily more capital-intensive in that it does not necessarily raise output per worker for the economy as a whole. This counter-intuitive result, at least by traditional neoclassical and Austrian teaching, obtains in Region 2.

 Region 2 Region 5 Economic Life of Machine Run for two years at higher rate of profits Run for one year at higher rate of profits Consumption per Person-Year Decreased at higher rates of profits Decreased at higher rate of profits Employment, given net corn output Increased at lower wage Increased at lower wage

6.0 Conclusion

Unambiguous physical measures are available here for examining central claims in Austrian capital theory. Net output in a stationary state in this model consists of a single consumption good, and labor is taken as homogeneous. Around a switch point exhibiting capital-reversing, a lower interest rate is associated with the adoption of a technique that produces less net output per unit of labor time. If a financial measure of capital intensity is adopted such that, around all switch points, lower interest rates are associated with the use of a technique with a longer average period of production, the association between more roundabout methods of production and greater productivity is broken. Austrian capital theory and Austrian business cycle theory fails to be sustained

In a model of fixed capital with only a single machine in use, the number of production cycles for the economic life of the machine is an unambiguous measure of roundaboutness. The numerical example in this chapter reveal that no connection necessarily exists between lower interest rates and the extension of the economic life of a machine. The example in Region 3 is a reswitching example. Necessarily, in a reswitching example in a model of fixed capital, a lower rate of profits is associated, around one of the switch points, with the truncation of the economic life of the machine.

The Austrian theory of capital is also refuted in Region 2 in the example, even with a single switch point. Around the switch point, a lower rate of profits is associated with a truncation of the economic life of a machine, not the extension of how long this machine is used in production cycles. And that truncation is associated with a more capital-intensive technique, as is seen in an increase in net output per worker. These examples demonstrate that the underlying vision behind the Austrian story is simply incorrect. Attempts to evade this proof by adopting a financial definition of capital simply ignore the points at issue.

Saturday, May 09, 2020

Financial Economics

This is a list of some of what I think one should know if one wants to talk to investors interested in theory. This post is not about making money and is probably not up-to-date. My references are fairly popular, and mostly old. I include one recent popular book as an example. Most of the references I do not recall very well, and I have not read Ben Graham. But many seem to know that Warren Buffet recommends this book. This post is non-critical. Keen and Quiggin in Debunking Economics and Zombie Economics, each have a chapter of criticism.

• Behavioral finance: The application of behavioral economics to finance.
• Beta: A parameter in the CAPM.
• Black Scholes formula: A formula for pricing options.
• Capital Asset Pricing Model (CAPM): A model that relates the risk of an asset to the market as a whole.
• Efficient Market Hypothesis (EMH): A model in which all information is quickly built into asset prices. The EMH comes in at least three types.
• Equity Premium Puzzle: The observed phenomena for stocks (or shares) to trade at higher prices, as compared to bonds, than can be justified by the EMH.
• Lévy distribution: A family of probability distributions that, except for the limiting case of the Gaussian distribution, have an infinite variance. The Cauchy distribution is also a member. Benoit Mandelbrot recommends this as a model for changes in asset prices.
• Martingale Theory: A branch of mathematics in which a stochastic process exhibits a special case of the Markov property. I recall learning about a drunkards walk and the gambler's ruin problem, but I do not recall this term in any of my formal math courses.
• Modigliani and Miller (M and M): A model that implies, under idealizations, that it does not matter if corporations finance investments with equity or debt.
• Noise trading: Trading on random variations in the price of an asset, instead of fundamentals. I know of this from some late 80s work of DeLong, Shleifer, Summers, and Waldmann.
• Stochastic Calculus, also known as Ito Calculus: A branch of mathematics in which one can talk about the derivatives and integrals of a set of random variables indexed on continuous time. Such a stochastic process is different from a single realization).
• Value-at-risk: A formula that applies to an investment portfolio.
• Volatility skew: An anomaly, inconsistent with the Black-Scholes formula, that emerged in markets for options.

One also needs to know about puts, calls, indices, credit default swaps, types of spreads (e.g. a broken wing butterfly spread) and so on if one wants to be a financial analyst. As usual, this is an aspirational post. I do not claim to know all of this, and maybe I have gotten some of the above incorrect.

References

Thursday, May 07, 2020

Elsewhere

• A podcast interview with Philip Mirowski about how, roughly, neoliberals are exploiting the Corona Virus crisis in America.
• A book, Nine Lives of Neoliberalism, edited by Dieter Plehwe, Quinn Slobodian, and Philip Mirowski. Somewhere this is or was freely downloadable in PDF.
• A podcast interview with Marshall Steinbaum about the Chicago school of economics.
• A podcast episode, by two economics professor, trying to present an overview of Joan Robinson.
• A downloadable book, Labour and Value: Rethinking Marx's Theory of Exploitation, by Ernesto Screpanti.
• Two young friends discuss Steve Keen's book, Debunking Economics.

Saturday, May 02, 2020

A Refutation Of Austrian Capital Theory And Austrian Business Cycle Theory

1.0 Introduction

I have not posted about a non-fluke switch point in a while. This is an example from Bertram Schefold. I have examined perturbations and variations of this example before.

Here I present an example with tables exhibiting arithmetic. Is this any more transparent than examples presented with graphs?

I have been listening to some lectures on YouTube, especially Richard Wolff. I now have another hypothesis why mainstream economists have been promoting lies, ignorance, and nonsense for half a century: fear. The history of economics includes purge after purge after purge. Maybe many mainstream economists are cowed by their rulers.

2.0 Technology

Three processes are available for use in production. Each process is specified by coefficients of production (Table 1), when operated at an unit level. The person-years of labor employed, the bushels of corn used up in a process, and the number of new and old machines are specified. Outputs consist of bushels corn and new and old machines. Corn is both a consumer good and functions as circulating capital. Machines function as fixed capital. A machine's productivity varies with its age. An older machine requires less labor to operate, but more circulating capital.

 Input Process (I) (II) (III) Labor 1/10 43/40 1 Corn 1/16 1/16 1/4 New Machines 0 1 0 Old Machines 0 0 1 Outputs Corn 0 1 1 New Machines 1 0 0 Old Machines 0 1 0

Suppose all three processes are run at unit level in parallel. At suppose at the start of the year, the firm has one new machine, one one-year old machine, and 3/8 bushels corn. The output of the first process is one new machine. The previously existing new machine is consumed in the second process, leaving an output of one one-year old machine. The third process uses up the one one-old year machine. So these processes reproduce the stock of new and old machines. But they also use up inputs of 3/8 bushels corn in producing a gross output of two bushels corn. Summing over all three processes, 2 7/40 person years of labor produce, with the capital stock, a net output of 1 5/8 bushels corn. The labor intensity of this technique is 87/65 person-years per bushel.

Assume free disposal for old machines. Another technique would be operated when the use of the machine is truncated after one year. In this case, one should consider the first and second processes being run in parallel at unit level, with a capital stock of one new machine and 1/8 bushels corn reproduced each year. Under this technique, 1 7/40 person years of labor are employed across the two processes. Net output is 7/8 bushels corn. The labor intensity is 47/35 person-years per bushel.

The more labor-intensive technique is the one with the truncated economic life of the machine. How many production cycles firms choose to run machines is an unambiguous physical measure in this example. And choosing to run the machine longer is a choice to adopt a less labor-intensive technique. Just as Eugen Böhm Bawerk says, a longer period of production, in some sense, is a more capital-intensive method. And that increase in the period of production results in greater output per worker. (I am not claiming that the number of production processes is the Austrian average period of production. I am merely noting how transparent the use of time is in this example.)

3.0 Some Accounting

One can easily see how a single firm would operate the last two processes in parallel, or only the process using the new machine, if the use of the machine is truncated. If such a firm was vertically integrated, they would also produce new machines with the first process. The firm can take the wage and the price of corn from the market, under various idealizations. For simplicity, I take a bushel corn as the numeraire. What prices would the accountants use to evaluate new and old machines?

3.1 At an Initial Wage

Consider a starting wage of 35/71 bushels per person-year and prices of machines shown in Table 2. For each process, the cost of capital inputs are the sum of the inputs of corn and machines of specified vintages, evaluated at the given prices. Wages are found from the labor input, evaluated at the given wage. Revenues are the sum of the outputs of corn and machines of specified vintages, evaluated at given prices. I assume wages are paid at the end of the year. So the rate of profits is the ratio of the difference between revenues and wages to the capital costs. With these prices, the owners are happy to operate all three processes. They make the same rate of processes in each, and prices do not signal that they should make any changes.

 w = 35/71 ≈ 0.4930, p0 = 99/568 ≈ 0.1743, p1 = 1/284 ≈ 0.003521 Capital Costs Wages Revenues Rate of Profits I 0.0625 0.04930 0.1743 100 percent II 0.2368 0.5299 1.004 100 percent III 0.2535 0.4930 1 100 percent

3.2 A Higher Wage

Now suppose the wage is 9,055/14,016 bushels per person-year. Table 3 shows accounting when the prices of new and old machines are unchanged. Notice that the rate of profits has fallen, with the rise in the wage, in all processes. But it has fallen to different levels, given that ratio of wages to the cost of capital originally varied among the processes. David Ricardo discusses this effect in the first chapter of his Principles of Political Economy and Taxation. Obviously, the price at which machines are entered into the firm's books must be changed to reflect the change in wages.

 w = 9,055/14,016 ≈ 0.6460, p0 ≈ 0.1743, p1 ≈ 0.003521 Capital Costs Wages Revenues Rate of Profits I 0.0625 0.06460 0.1743 75.5 percent II 0.2368 0.6945 1.004 30.5 percent III 0.2535 0.6460 1 39.6 percent

Table 4 shows a set of prices such that the same rate of profits is obtained in all three processes. This is not the end of the story, though, The price of a one-year old machine is slightly negative. (My approach for perturbing reswitching examples can make this price more noticeably negative, but maybe I would end up with even more messy fractions.) Instead of decreasing the revenues for the second process from the output of old machines, the managers of the firm can simply throw the old machine away and enter a price of zero for it on its books.

 w = 9,055/14,016 ≈ 0.6460, p0 ≈ 0.1531, p1 ≈ -9.563 x 10-5 Capital Costs Wages Revenues Rate of Profits I 0.0625 0.06460 0.08852 41.64 percent II 0.2156 0.6945 0.3054 41.64 percent III 0.2499 0.6460 0.3539 41.64 percent

Table 5 shows the result of truncating the use of machines. One must, however, set a new accounting price for new machines, as well. Without this adjustment, different rates of profits are obtained in the first two processes. Notice the rate of profits is indeed lower in the third process, which is not used by cost-minimizing firms.

 w = 9,055/14,016 ≈ 0.6460, p0 ≈ 0.15313, p1 = 0 Capital Costs Wages Revenues Rate of Profits I 0.0625 0.06460 0.15313 41.64 percent II 0.21563 0.6945 1 41.68 percent III 0.25 0.6460 1 41.58 percent

Table 6 shows the final results of correct accounting, with increased wages. The rate of profits, r, is 5/12, halfway between 1/3 and 1/2. Those rates of profits are the switch points in this example. I have told this story as a matter of accounting for vertically integrated firms. But these are the only prices on the market consistent with a long period position with the postulated wage. How and whether market prices would converge to these prices of production, in a gravitational process (as in Adam Smith's metaphor) is not clear to me. I like the idea that Sraffa's book is about little more than accounting. This idea is not too far away from what Ajit Sinha has been arguing for a number of years.

 w = 9,055/14,016 ≈ 0.6460, p0 = 1,431/9,344 ≈ 0.15315, p1 = 0 Capital Costs Wages Revenues Rate of Profits I 0.0625 0.0646 0.15315 41.67 percent II 0.21565 0.6945 1 41.67 percent III 0.25 0.6460 1 41.58 percent

3.3 Summary

 Wage Labor Intensity Rate of Profits 35/71 ≈ 0.4930 bushels per person-yr. 87/65 ≈ 1.338 person-yrs. per bushel 100 percent 9,055/14,016 ≈ 0.6460 bushels per person-yr. 47/35 ≈ 1.343 person-yrs. per bushel 5/12 ≈ 41.7 percent

At a higher wage, firms want to run machines for one year, not two years. The economic life of machines is shortened from the physical life. And firms want to hire more workers to produce a net output.

Why might the wage rise and the rate of profits fall? In one theory, known to be nonsense, a shock might lead to labor be less abundant, as compared to capital. Perhaps people become more forward-looking and more willing to save, or the population falls for some reason. Wages rise, and firms take this as a signal to substitute capital for labor. But, in the example, a higher wage is associated with the adoption of a less capital-intensive technique and a rise in labor intensity. Somehow, if equilibrium is to be obtained, an increase in the labor force must be accompanied by a rise in wages.

Perhaps one can find a financial measure of capital, such as Hicks' average period of production, where a higher capital-intensity is always associated with a lower rate of profits around a switch point. But then one would have to grapple with the fact that more 'capital' does not always result in more output. Austrian school economists cannot seem to handle that their theory of 'malinvestment' is just incorrect.

Saturday, April 25, 2020

Old Findings For New Times

From John Kenneth Galbraith's The Affluent Society (1958), I know that widespread attitude to waged work among some in the United States is outdated. The conventional wisdom is that work is necessary because it is needed to produce the goods that sustain society. That was largely true before productivity increased so much, for example, in the post war golden age. Some jobs are still essential, by any narrow definition, but much wage work goes to creating goods and services that in any previous society would have been considered useless luxuries. (As I get older though, I disagree with Galbraith about medicines to improve peristalsis.) But waged work, in the current system, is essential for providing the income needed to sustain the demand to keep the system going.

From Paul Davidson and Joan Robinson, I know about the distinction between historical and logical time. It as not as if the economy is in an equilibrium that will be approached again, and quickly, after a downward shock is removed. People will recall, and those who are out of work will try to cut back their spending. States and localities that have had to increase their spending to address such a shock will need to retain the ability to spend, and even to increase spending.

From Michal Kalecki's "Political aspects of full employment" (1943), I know that the ruling bourgeois class cannot be counted on to support what is even in their immediate short term financial interest. Unemployment is convenient for keeping the mass majority of the population, the labor force, cringing and hard to press for more of a share in the commodities that they produce. Even so, the backwardness of politics in the United States these days is hard to explain.

From Hannah Arendt's Eichmann in Jerusalem (1963), I know that one strategy in implementing totalitarianism is to declare a marginal group stateless, outside your laws and international laws. Refugees and immigrants are one such group that one could try to apply this strategy to, if you are for evil. From Carl Schmitt's The Concept of the Political (1932), I know that sovereign is he who can declare the state of exception. I resent that these ideas are relevant today.

Thursday, April 16, 2020

A Fluke Switch Point On The Wage Frontier

 Figure 1: A Switch Point On The Wage Frontier with Wage Curves Tangent

This post extends the example in my article in Structural Change and Economic Dynamics, suitably emended.

Figure 2 shows an enlargement of part of the parameter space. The parameters of the point where the boundaries of Regions 1, 2, and 3 intersect are shown. In Region 1, the Beta technique is uniquely cost-minimizing for all feasible wages; there are no switch points. Region 2 is a reswitching example, with Beta cost-minimizing at low and high wages. One switch point exists in Region 3, with the Beta technique cost-minimizing at low wages.

 Figure 2: Part of Parameter Space

The boundary between Regions 1 and 2 is tangent to the boundary between Regions 1 or 2 and Region 3. In what I am calling a reswitching pattern, the scale factor for the rate of profits is found as a double root for a polynomial equation. I have extended the boundary between Regions 1 and 2, with the dotted line, to show where this double root occurs with a negative scale factor for the rate of profits.

Figure 1, at the top of this post, shows wage curves for the example for the parameter values noted in Figure 2. (I used Octave to help with my arithmetic.) The Beta technique is cost-minimizing for all feasible wages. When the wage is at its maximum, the Alpha technique is also cost minimizing. And, at the switch point, the two wage curves are tangent. This is a fluke switch point twice over.

This sort of fluke switch point cannot be expected to be found in empirical data from, say, National Income and Product Accounts. The fluke cases I have been developing are important in that they illustrate partitions in a parameter space for certain models of the choice of technique. They arise as certain characteristics of markets vary with a perturbation of model parameters.

I am not sure what to make of structures within the parts of the parameter space I have been exploring. If you think about, you can see why that must be a point of tangency in Figure 2. Maybe the most striking structure I have found is parallel lines for partitioning a parameter space associated with an example of Harrod-neutral technical change.

I write this stuff as escapism. A lot of economists I build on are in Italy, which is a worrisome place to be these days. I hope you are doing well.

Thursday, April 09, 2020

A Fluke Case With Two Fluke Switch Points

 Figure 1: Switch Points On The Axis For The Rate Of Profits And At r = -100 Percent

This is an example of a fluke case in the analysis of the choice of technique. The interest in flukes, for me, is that they show how the characteristics of markets can change. They provide insight into structural economic dynamics, as Luigi Pasinetti calls it.

I have previously shown a fluke case, with a switch point on the axis for the rate of profits with a real Wicksell effect of zero. A perturbation of the example can lead to a reswitching example. The switch point at a wage of zero (when the workers live on air) then becomes one at a positive wage. And around that switch point, a higher wage is associated with cost minimizing firms hiring more workers to produce a given net output.

In the example in this post, the switch point on the axis for the rate of profits exhibits neither a forward nor a reverse substitution of labor. The labor coefficient in the corn industry does not vary with the processes in the technique. The Alpha technique has a ghostly presence. It can only be chosen, and not even uniquely so, when the wage is zero. A perturbation of this example can lead to one of the reverse substitution of labor. The switch point on the axis for the rate of zero would also become one at a positive wage. And that switch point might be the only switch point on the frontier at a non-negative rate of profits. Around that switch point, a higher wage is associated with cost-minimizing firms hiring more workers to produce a given gross output of corn. The labor coefficient in the corn-producing process for the technique preferred at a higher wage is larger.

 Input Iron Corn Industry Alpha Beta Labor 1 0.64097922 0.64097922 Iron 9/20 0.00157618 0.01686787 Corn 2 0.48125981 0.0674715

Table 1 specifies the technology in my usual way. I assume labor is advanced, and wages are paid out of the product at the end of a production cycle. I take a unit of corn as numeraire. Prices of production are here defined with a uniform rate of profits between the industry. I found this example with numerical exploration, so there is some round-off error in the figures.

This post is another demonstration that explaining wages and employment by supply and demand, even under ideal competitive conditions, is incoherent nonsense.

Friday, April 03, 2020

A Market Algorithm

 Figure 1: Specification of a Market Algorithm
1.0 Introduction

An approach to the analysis of the choice of technique, in keeping with construction of the outer envelope of wage curves, is to consider replacing processes, more or less, one at a time. This post presents this approach as following an algorithm.

Assume that a set of techniques exist where all techniques are at least viable, indecomposable, and produce the same set of commodities. From the set of techniques, one can form a set of processes. In each process, workers produce a single commodity at the end of the year from certain inputs. The inputs, by assumption, are totally consumed in the course of the year. I also assume that the numeraire is specified.

Consider the algorithm specified by the flowchart in Figure 1. For this to be an algorithm, Steps 1 and 3 must be fully specified. One might as well assume that a known pseudo random number generator is used with a specified initial seed. Whether or not a candidate process yields extra profits is found in Step 5 with the prices of production calculated in Step 2. A process yields extra profits if and only if:

p a.,j (1 + r) + w a0,j > pj

where a0,j is the direct labor coefficient, and a.,j is a column vector for the new process. I am imagining that a.,j is the j column of the Leontief input-output matrix for a new technique. This new technique is formed by replacing a process in the technique previously selected in Step 2. Since, by assumption, no joint production exists, the process to be replaced is easily found. It is the process in the current technique that produces the same commodity as the candidate process. I have taken the wage as given in this specification of the algorithm. One could just as well take rates of profits as given.

This algorithm converges to a cost-minimizing technique. Consider the sequence of Steps enumerated as ‘2, 3, 4, (5, 7, 9)*, 5, 6, 2’. This expression denotes a single path around the loop on the bottom left of Figure 1, including zero or more paths around the loop on the right. As long as the loop on the right is repeated less times than the number of techniques, this path can be repeated. The question arises whether or not this algorithm contains an infinite loop. In a simple case, a process would be introduced into a technique because it is cost-minimizing for prices corresponding to the first technique, and that first technique would be cost-minimizing at the prices corresponding to the new technique. It can be shown that the existence of an infinite loop is impossible, under the assumption that no joint production exists. The algorithm always terminates.

(The use of metalinguistic symbols of parentheses and an asterisk to denote a repeated sequence of symbols is a convention in defining regular expressions. A sequence of symbols in a language, where the grammar of that language is specified by a regular expression, is accepted by a finite state machine, a type of automata. This is the lowest level of the Chomsky hierarchy. Chomsky (1965) uses transformational grammars to characterize human languages, which he argues are at the highest level of the hierarchy.)

Furthermore, except at switch points, the cost-minimizing technique found by the algorithm is unique. Which technique is initially selected at Step 1 or how processes are ordered at Step 3 does not matter, except for performance. The same cost-minimizing technique is ultimately found. The algorithm terminates with the selection of any one of the techniques that are cost minimizing at a switch point, depending on these details.

The algorithm is specified sequentially in Figure 1. Steps 3, 4, 5, 7, and 9 can be distributed. Inasmuch as this algorithm is executed in a capitalist economy, these steps are, in fact, distributed across firms. One might also modify the algorithm to apply when the set of processes and techniques are not known at that start of algorithm. Innovation and technical progress can be accommodated with an appropriate modification of Step 4 and Step 9. Step 7 should be eliminated, and the algorithm would be defined without a termination step, like daemons in operating systems. When the algorithm is modified for distributed processing, more than one process might be introduced into a technique simultaneously, including in the same industry. For which technique, then, are prices calculated in Step 2? This relates to the question of when labor expended in new processes is ‘socially necessary’, as Marx put it.

References
• Bidard, Christian. 2004. Prices, Reproduction, Scarcity. Cambridge: Cambridge University Press.
• Chomsky, Noam. 1965. Aspects of the Theory of Syntax. Cambridge: M.I.T. Press.

Saturday, March 28, 2020

Another Example Of The Factor Price Frontier In The Space Of Rental Prices

 Figure 1: Real Factor Price Frontier
1.0 Introduction

I am planning on posting at an even slower rate for a while.

This post continues my exploration of a way of visualizing the choice of technique proposed by Carlo Milana. In this post, I show how to correctly apply his visualization to an example from my 2017 ROPE paper.

2.0 Technology

Two techniques are available for producing corn, the consumption good. Each technique consists of a flow-input, point-output technology, with a finite number of dated labor inputs. The technology can also be represented with intermediate produced commodities explicitly shown. Table 1 sets out the example as a Leontief input-output table, in some sense. Each column lists the inputs in a production process needed to produce a unit output of the commodity named in the column heading. The entries for the row for corn are all zero, indicating it is not an input into any production process. All processes require a year to complete and completely use up their commodity inputs in producing their outputs. Each process requires a year to complete, and inputs must be hired at the start of the year, independently of when payments for inputs are contracted to be paid out. The example includes four factors of production: labor, and three capital goods.

 Input Industry 1 Industry 2 Industry 3 Corn Industry Alpha Beta Labor 50 0 115 66 0 Commodity 1 0 1 0 0 0 Commodity 2 0 0 0 1 0 Commodity 3 0 0 0 0 1 Corn 0 0 0 0 0

The Alpha process consists of the processes for producing the first two commodities and the corn-producing process labeled Alpha. The Beta process consists of the process for producing the third commodity and the corn-producing process labeled Beta. The net output of a technique, when all processes comprising that technique are operated at the unit level, is a bushel of corn. The non-labor inputs needed to continue production with that technique are reproduced in the same quantities with this activation of processes.

3.0 Price System

I here formulate systems of equations (and inequalities) in terms of spot prices and rental prices for factors of production. A rental price pays for the services of a factor of production, and is paid at the end of the year. I formulate the example with the following variables:

• wL: The wage, paid at the end of the year. Also known as the rental price for labor.
• p1: The spot price for the first produced capital good.
• w1: The rental price for the services of the first capital good.
• p2: The spot price for the second produced capital good.
• w2: The rental price for the services of the second capital good.
• p3: The spot price for the third produced capital good.
• w3: The rental price for the services of the third capital good.
• pc: The spot price for corn.
• r: The interest rate.

In a circulating capital model of long period positions, rental prices and spot prices relate like so:

wj = pj(1 + r), j = 1, 2, 3

Since this is not a model of a slave economy, no spot price exists for labor. Corn is never used as a factor of production and does not have a rental price.

The price equations are such that no extra profits can be made in any process. Furthermore, costs do not exceed revenues in any operated process. Corn is assumed to be the numeraire. Figure 1, above, graphs the solution. This is actually a two-dimensional representation of a projection of a four-dimensional structure into three dimensions. The fourth dimension is the rental price for the first produced capital good.

3.1 Prices for the Alpha Technique

Suppose managers of firms have adopted the Alpha technique. The condition that the corn-producing process in the Alpha technique is adopted and pays no extra profits is expressed as:

66 (wL/pc) + (w2/pc) = 1

The above equation gives rise to the blue plane in the figure. The analogous condition for the process to produce the first capital good is expressed as:

50 (wL/pc) = (p1/pc)

The same sort of condition for the process for producing the second capital good is:

(w1/pc) = (p2/pc)

When the Alpha technique is adopted, prices may or may not enable managers of firms to activate the process for producing the third capital good. This condition is express as an inequality:

115 (wL/pc) ≥ (p3/pc)

In the graphed solution, the above inequality is treated as an equality, giving rise to the factor-price curve for the Alpha technique. An inequality arises for the corn-producing process in the Beta technique.

(w3/pc) ≥ 1

The above inequality cannot be combined with the other equations in the Alpha price system except between the switch points. That is, the factor price curve for the Alpha technique is the factor price frontier only between the switch points.

3.1 Prices for the Beta Technique

Now suppose that managers of firms have adopted the Beta technique. The condition that the corn-producing process in the Beta technique is adopted and pays no extra profits is expressed as:

(w3/pc) = 1

The above equation gives rise to the red plane in the figure. The analogous condition for the process to produce the third capital good is expressed as:

115 (wL/pc) = (p3/pc)

Inequalities arise for the processes for producing the first and second capital goods:

50 (wL/pc) ≥ (p1/pc)

(w1/pc) ≥ (p2/pc)

Likewise, an inequality arises for the corn-producing process in the Beta technique:

66 (wL/pc) + (w2/pc) ≥ 1

3.3 Solution of Price Equations

Table 2 displays the solution for the two systems of equations and inequalities. The equations impose a relationship between the wage and the interest rate, either of which could be given from outside the period of production. Spot prices for the three capital goods can be found by discounting rental prices to the start of the year.

 Price Alpha Beta (wL/pc) 1/(2 (33 + 25 (1 + r)2)) 1/(115 (1 + r)) (w1/pc) 25 (1 + r)/(33 + 25 (1 + r)2) 10/23 (w2/pc) 25 (1 + r)2/(33 + 25 (1 + r)2) 10 (1 + r)/23 (w3/pc) 115 (1 + r)/(2 (33 + 25 (1 + r)2)) 1

In both systems of equations, the rental prices of factors of production are expressed as functions of the interest rate. That is, factor price curves in Figure 1 are specified parametrically. Both factor price curves lie in their respective planes. Switch points lie on the intersection of the two planes in Figure 1. Outside the switch points, the factor price curve for the Beta technique is the factor price frontier. Between the switch points, the factor price curve for the Alpha technique is the factor price frontier.

4.0 Conclusion

Milana is correct in asserting that the interest rate is not a factor price, that is, a rental price for the services of a factor of production. The label "factor price frontier" should properly be reserved for a locus in the space of rental prices. This post has illustrated these claims by showing how to draw such a factor price frontier for a reswitching example with a specific structure.

Saturday, March 14, 2020

On "Democratic Socialism"

1.0 Introduction

This post is about some usages of the phrase "democratic socialism".

2.0 Democratic Socialists of America (DSA)

In the context of current American politics, I think the most salient usage today of this phrase is associated with Bernie Sanders' campaign and with the Democratic Socialists of America.

DSA was founded in 1982 when the Democratic Socialist Organizing Committee (DSOC) merged with the New American Movement (NAM). The DSOC was established with Michael Harrington leading others out of the Socialist Party (SP) (Gorman 1995: p. 144-145). Apparently, 500 people attended the DSOC founding convention in 1973 (Harrington 1988: 17).

Michael Harrington will forever be known for The Other America. In this book, he deliberately did not use the word "socialism". And this book had some influence on the policies of Kennedy and Johnson and the latter's war on poverty. He also was something of a theorist. The Twilight of Capitalist is an attempt to apply a tradition of an "underground" Marxism to the conjuncture of 1970s "late capitalism". Harrington mentions Rosa Luxemburg, George Lukacs, Karl Korsch, the Frankfurt school, and Antonio Gramsci, for example, as well as more current thinkers. Hegel and dialectics are important to how Harrington understood Marx.

Some of the left want to establish a political party for the working class and saw that the Democratic party is not such a party. So they think they should have their own party. DSA is and DSOC was the opposite of that. DSA can be said to be openly pursuing a strategy of entryism. The goal is to influence the Democrats from within.

3.0 An Older Usage

But the term goes back much further than these movements in the United States in the 1970s and 1980s. "Democratic socialism" was used in English, in the United States, in the 1950s and early 1960s, by scholars discussing the "revisionism" of Eduard Bernstein.

I turn to the Second International shortly before 1900. As I understand it, a literal translation of the leading socialist party was something like the "Social Democratic Party of Germany" (SPD). I do not want to read much into the phrase "social democratic" here. At the time, Georgi Plekhanov and Vladimir Lenin were in the Russian Social Democratic Party.

In 1899, Engel's literary heir, Eduard Bernstein published a book. The Preconditions of Socialism and the Task of Social Democracy is a literal translation of its title, I guess. The book built on articles he published earlier in Die Neue Zeit. "The final goal is nothing to me, the movement is everything" is a well-known pithy statement from Berstein. He argued for pursuing reforms, supporting trade unions, and a parliamentary party on the foundation of universal suffrage.

Rosa Luxemburg saw a chance here to raise her profile in the socialist movement. Her response, Reform or Revolution, is another classic.

I have always thought of Karl Kautsky as a politician trying to keep advocates of all these tendencies within a single party. His grandson (Kautsky 1994) argues he was consistent, from the time that Lenin hailed him as the leader of the orthodoxy to the time Lenin called him a renegade. Here is a quote from the later time:

"For us... Socialism without democracy is unthinkable. We understand by modern Socialism not merely social organisation of production, but democratic organisation of society as well. Accordingly, Socialism is for us inseparably connected with democracy. No Socialism without democracy." -- Karl Kautsky, The Dictatorship of the Proletariat (1918), as quoted in Kautsky (1994).

By the way, John Kautsky tells of running into college students who were surprised that his granddad's first name was Karl; they thought somehow or other it was "Renegade"

Around 1900, splits and disagreements between more and less radical factions in socialist parties were an international phenomenon. In France, the syndicalist Georges Sorel was a radical and Jean Jaurès was more reformist. The Russians split in 1903, between the Mensheviks and the Bolsheviks. I gather a large part of the Menshevik faction walked out of the party congress, leaving the Bolsheviks the ability to call themselves the "majority", even though they were a minority of the party.

I skip forward any number of years. Maybe half of Sidney Hook's 1955 book is short selections from others. His reading number 54 is excerpted from a statement "adopted by the Socialist International at Frankfurt-on-Main, Germany, 1951." And it is entitled, "The aims and tasks of democratic socialism".

I think of Sidney Hook as an exemplar of an advocate of social democracy. I guess this section argues that the distinction between social democracy and democratic socialism was, at one time, not clear.

References
• Eduard Bernstein. 1961. Evolutionary Socialism: The Classic Statement of Democratic Socialism (trans. by Edith C. Harvey). Schocken.
• Peter Gay. 1952. The Dilemma of Democratic Socialism. Columbia University Press.
• Robert A. Gorman. 1995. Michael Harrington: Speaking American. Routledge.
• Michael Harrington. 1962. The Other America: Poverty in the United States.
• Michael Harrington. 1976. The Twilight of Capitalism. Simon and Schuster.
• Michael Harrington. 1988. The Long-Distance Runner: An Autobiography. Henry Holt.
• Sidney Hook. 1955. Marx and the Marxists. D. Van Nostrand.
• Maurice Isserman. 2001. The Other American: The Life of Michael Harrington.
• John H. Kautsky. 1994. Karl Kautsky: Marxism, Revolution & Democracy. Transaction Publishers.

Saturday, March 07, 2020

Two Propositions: Neoclassical Economics Is Incoherent; A Classical Theory Of Value Exists

 Some Mathematics Useful In Understanding Classical Political Economy

I consider the following to have been well established about half a century ago:

1. Marginalism or so-called neoclassical economics is impossible to formulate consistently and with practical applications.
2. A mathematically rigorous approach to the classical theory of value, from William Petty through David Ricardo, including Karl Marx, exists.

Mainstream economists ignore the truth of both propositions. Until they stop spouting lies and nonsense, these propositions should be re-iterated again and again. (On the other hand, I appreciate the work involved in compiling National Income and Products Accounts.)

I find a difficulty in publishing re-iterations of these propositions. I expect editors and reviewers of, say, the American Economic Review, the Journal of Political Economy, or the Quarterly Journal of Economics would simply not publish papers stating either proposition. You can publish articles in, say, the Review of Political Economy which re-state these propositions, but any such article must contain something novel. For my purposes, I seem to stumbled upon a research program that includes:

• Exploring what still holds when prices of production are defined with persisting non-uniform rates of profits.
• Exploring and visualizing the effects of perturbing parameters in models of prices of production.
• Refuting those who claim to have some other analysis of the choice of technique in which, say, reswitching examples are supposedly mistaken.

Others have other focii for their research. For example, historians of political economy might be interested in the publication of critical editions of Marx-Engels Collected Works (MEGA) or of Piero Sraffa's archives. Over the last couple of decades, some have worked on exploring problems in joint production and some limitations of the long period method, such as non-reproducible, exhaustible natural resources (as seen, for example, in the corn-guano model). Many other issues have been explored.

Sunday, February 23, 2020

Update To Example In Vienneau (2019)

Maybe this post should be titled "Erratum" or "corrigendum". I have an example in my paper last year in which wage frontiers are supposed to vary with two parameters. One is the markup in the "iron" industry. And the other is σ t. The example should be as in Table 1. All the theory and the visualizations in the paper work out with this example.

 Input Industry Iron Corn Alpha Beta Labor a0,1 = 1 aα,0,2(t) = (5191/5770) e(1/10) - σ t aβ,0,2 = 305/494 Iron a1,1 = 9/20 aα,1,2(t) = (1/40) e(1/10) - σ t aβ,1,2 = 3/1976 Corn a2,1 = 2 aα,2,2(t) = (1/10) e(1/10) - σ t aβ,2,2 = 229/494

Thursday, February 20, 2020

Elsewhere

• John Weeks presents Joan Robinson's contributions to the Cambridge Capital Controversy (CCC).
• A discussion, from 2006, on Daily Kos, about one of my attempts to explain the CCC.
• A post, from 2016, on Naked Capitalism about how the CCC shows microeconomics is all wet.
• J. W. Mason has a handout explaining a definition of capital.
• Doyne Farmer, Fotini Markopoulou, Eric Beinhocker, and Steen Rasmussen, in an essay in Aeon, Collaborators in creation, provide an overview of complexity economics.
• An overview, from 2018, about how women were deliberately kicked out of software development in Great Britain.

Saturday, February 15, 2020

Universal Basic Income: Some Advocates And Analysts

"In fact, the realm of freedom actually begins only where labour which is determined by necessity and mundane considerations ceases; thus in the very nature of things it lies beyond the sphere of actual material production. Just as the savage must wrestle with Nature to satisfy his wants, to maintain and reproduce life, so must civilised man, and he must do so in all social formations and under all possible modes of production. With his development this realm of physical necessity expands as a result of his wants; but, at the same time, the forces of production which satisfy these wants also increase. Freedom in this field can only consist in socialised man, the associated producers, rationally regulating their interchange with Nature, bringing it under their common control, instead of being ruled by it as by the blind forces of Nature; and achieving this with the least expenditure of energy and under conditions most favourable to, and worthy of, their human nature. But it nonetheless still remains a realm of necessity. Beyond it begins that development of human energy which is an end in itself, the true realm of freedom, which, however, can blossom forth only with this realm of necessity as its basis. The shortening of the working-day is its basic prerequisite." -- Karl Marx, Capital, Volume 3, Chapter 48.

This post, I guess, is somewhat about current events. I claim no comprehensiveness for my impressions on the topic. I find my references are aspirational more than usual; some do not even seem to be in print. In this post, I do not compare and contrast UBI with a Job Guarantee, which seems to be argued about a lot on twitter.

Three properties define a UBI, according to Philippe Van Parijs. It is paid to individuals, not households. One obtains a UBI payment independent of need or any other sources of income. And it is not conditional on one's willingness to accept a job or on a requirement to have worked in the past or to work in the future.

I think a UBI is consistent with Keynes' vision in 1931 of "Economic possibilities for our grandchildren". Whether increases in productivity are broadly shared, including in increased leisure or non-work time seems not to be determined by technology. In some sense, the use of such progress for progressive ends is a result of collective choice. By the way, increased non-work time is part of a response to the global climate crisis. I have seen statistics about how less carbon is emitted during recessions. If we labor less voluntarily, and share time somewhat fairly, those trends would continue, I gather. So UBI could be one component of a green policy.

I always like to find those on the right stating their principles support supposed policies of the left. I find Hayek's ideas in The Road to Serfdom as inconsistent with UBI:

"There is no reason why in a society which has reached the general level of wealth which ours has attained the first kind of security should not be guaranteed to all without endangering general freedom. There are difficult questions about the precise standard which should thus be assured; there is particularly the difficult question whether those who thus rely on the community should indefinitely enjoy all the same liberties as the rest. An incautious handling of these questions might well cause serious and perhaps even dangerous political problems; but there can be no doubt that some minimum of food, shelter, and clothing, sufficient to preserve health and the capacity to work, can be assured to everybody." -- Hayek (1944, Chapter IX).

Hayek argues for too much, in that UBI proponents do not claim to be assuring, by this policy alone, that some necessary minimum will be achieved. He argues for too little, in that proponents of UBI do not want to even consider limiting the liberties of those who receive it. Nevertheless, the above quotation, and the chapter in which it is in, might be of interest to advocates of a UBI. From Hayek's perspective, a UBI does not seem to be a threat to the liberty of private property.

I have been reading Steve Wright about a much more radical movement that developed ideas close to a UBI. His book is an intellectual history of operaismo (workerism) and the area of autonomy in Italy in the 1960s and 1970s. Mario Tronti and Antonio Negri are two leaders of these tendencies, albeit they seem like the kind that are, in some sense, leaderless and spontaneous. I find some of the slogans and practices (for example, "the refusal of work" and "self reduction" (autoriduzione)) associated with autonomism intriguing. Workerists developed the concept of the social worker, the idea that the self-reproduction of the economy occurred not only in factories, but outside including by such non-waged individuals as students and housewives. The autonomists wanted to separate pay from productive labor, including a wage for housework. This social "wage" sounds a lot like an UBI to me.

I now turn to an author I know even less about. Philippe Van Parijs seems to be the most prominent academic advocate of UBI. He comes out of the tradition of analytical Marxism. I associate G. A. Cohen, Jon Elster, and John Roemer with this approach, although I knew that the September group had more members. Here Chris Bertram interviews Van Parijs. Van Parijs seems to have a number of books, more than I list below, in which others comment in his idea for UBI and in which he responds. Apparently, in discussing the impact of an UBI on work ethic, this literature also turns to a stereotype of surfers. (I was under the impression that the sunfish sailboat was also the product of a beach bum.)

My favorite approach to economics emphasizes questions of viability and what is needed to sustain human societies. Guglielmo Chiodi connects Sraffa's book with an UBI and normative concerns. No physical surplus is produced in the model in the first chapter of Sraffa's book. The inputs of production processes can be just reproduced from the outputs. Prices of production are determined by the need to redistribute these outputs, in accordance with a division of labor. One can consider these inputs as including commodities to sustain workers, just as they might include feed for horses. I think I take from Bertram Schefold the idea that the inputs might also include investment goods and capitalist consumption. Chiodi goes further. He suggests inputs include consumption by those who are neither capitalists nor working. And he reads Sraffa as emphasizing non-market institutions and non-market values. I am not sure I agree with this reading, but I find it of interest. I agree with Chiodi that Sraffa is more than an internal critique of neoclassical economics, but points to an alternative approach to economics. I do not think I have read anybody other Chiodi as connecting Sraffa to UBI. But I still have more to learn about the autonomist slogan of "the wage as the independent variable."

References

Saturday, February 08, 2020

Reswitching With Markup Pricing And Fixed Capital

 Figure 1: Two Dimensional Pattern Diagram
1.0 Introduction

This post extends an example from Bertram Schefold. It presents markup pricing in an example with a machine that can be operated for two years or junked after one year. This is a case of joint production in which, unlike in some cases, the choice of technique can still be analyzed by the construction of the wage frontier. Also, I do not think the question of requirements for use enter in here, and all matrices are square. As usual, this is a proof that "the marginal productivity theory of distribution" (and the neoclassical theory of supply and demand) "is all bosh" (Robinson 1961).

2.0 Technology

Table 1 shows the coefficients of production for the three processes comprising the available technology. Inputs must be available at the beginning of the year, and outputs become available at the harvest at the end. In the first process, labor uses inputs of corn to produce a new machine. That machine is used by labor in the second process, with inputs of seed corn, to produce more corn and a one-year old machine. In the third process, labor uses inputs of seed corn and the one-year old machine to produce corn. (I did think of calling the machine a "tractor".) The machine varies in physical efficiency over the course of its lifetime.

 Input Machine Industry Corn Industry One Process Another Process Labor a0, 1 = 1/10 a0, 2 = 43/40 a0, 3 = 1 Corn a1, 1 = 1/16 a1, 2 = 1/16 a1, 3 = 1/4 New Machines a2, 1 = 0 a2, 2 = 1 a2, 3 = 0 Old Machines a3, 1 = 0 a3, 2 = 0 a3, 3 = 1 Output Corn b1, 1 = 0 b1, 2 = 1 b1, 3 = 1 New Machines b2, 1 = 1 b2, 2 = 0 b2, 3 = 0 Old Machines b3, 1 = 0 b3, 2 = 1 b3, 3 = 0

The technology is summarized by a row vector a0, the input vector A, and the output vector B. The example satisfies various assumptions that show the economy hangs together, in some sense, and that it is more than viable.

The technology also presents a choice of technique. Managers of firms in corn-production may want to truncate the use of the machine to one year, given certain configurations of prices. In the Alpha technique, the machine is used for two years. I call the technique in which the machine is only used for one year the Beta technique.

3.0 Prices of Production

The price equations for the Alpha technique are:

p A (I + r S) + w a0 = p B

where p is a row vector of three prices (the price of corn, the price of a new machine, and the price of a one-year old machine), w is the wage, and r is the scale factor for the rate of profits. In a common notation, I is the identity matrix. The square matrix S is a diagonal matrix. Its diagonal elements express persistent differences in the rate of profits among processes or industries. The rate of profits in the jth process is r sj.

I take corn as the numeraire. This can be expressed as:

p e1 = 1

where e1 is the first column of the identity matrix.

4.0 Some Visualizations of the Solution

Assume, without loss of generality, that the markup coefficient, s1, in the process producing new machines is unity. Figure 2 shows how the choice of technique varies with the wage in the case where the markup coefficients in corn-producing processes do not vary with the age of the machine. When barriers to entry in producing new machines ensure that the markups in corn production fall appreciably below the overall rate of profits, managers of corn-producing firms will operate the machine for its full physical life, whatever the wage. On the other hand, roughly, if the corn-producing industry maintains barrier to entries, the machine will be operated for its full life only at low and high wages. At intermediate wages, the use of the machine will be truncated after one year. As still higher markups in corn-production, the machine will only be operated for the full two years for high wages.

 Figure 2: A Pattern Diagram

Figure 2 can be constructed in two ways. One is by constructing the wage frontier out of the wage curves for the two techniques. Figure 3 shows an example. This case is for what I call a reswitching pattern. The two wage curves are tangent at the single switch point. In finding the wage curve for the Alpha technique, one can also solve for the price of a one-year old machine. In the analysis of truncation with fixed capital, the machine is operated for only one year when this price turns negative. Switch points between the two techniques arise for wages in which the price of a one-year old machine is zero. I believe this analysis applies with the formulation of markup pricing in this post.

 Figure 3: Wage Curves for a Reswitching Pattern

Figure 1, at the top of this post, generalizes the analysis to all values of the markups in the corn-producing processes. Regions are indicated in which the machine is operated for two years, whatever the wage; in which this is an example of reswitching; and in which the machine is operated for two years only for high wages. The dashed (45-degree) line shows the case in which the markup is the same in both corn-producing processes.

I wonder if it makes any economic sense to consider cases off the 45-degree line in Figure 1. In this simple example, the two corn-producing processes are in the same industry, in some sense. If one agrees with this limitation for economic sense, a question arises. How, in some formulation of markup pricing, should such constraints be formulated, in general, for prices of production in models of joint production? Could markups, for instance, vary between the production of mainly wool and mainly mutton? Since definitions of even basic commodities vary among analyses of joint production, I do not see how to identify such processes in general where you might want to raise the question. Maybe these questions could be partly addressed by considering the process of vertical integration.

5.0 Conclusions

This post has illustrated that the analysis of the choice of technique must be performed in models with fixed capital. Managers of firms always have the choice of truncation, of adopting a technique in which the economic life of a machine is shorter than its physical life. This presents a challenge to attempts to justify Marx's theory of value with Sraffa's standard commodity. I have not even gone into some of the complications raised by pure joint production and models in which multiple types of machines are used.

References
• Luigi L. Pasinetti (ed.) 1980. Essays on the Theory of Joint Production. New York: Columbia University Press.
• Joan Robinson. 1961. Prelude to a critique of economic theory. Prelude to a critique of economic theory 13: 53-58.
• Bertram Schefold. 1980. Fixed capital as a joint product and the analysis of accumulation with different forms of technical progress. In Pasinetti 1980.