Monday, January 15, 2018

Start of a Catalogue of Flukes of Fluke Switch Points

I claim that the pattern analysis I have defined can be used to generate additional fluke switch points. I am particularly interested in switch points that are flukes in more than one way (local patterns of co-dimension higher than one) and fluke switch points that are combined with other fluke switch points or some aspect of other switch points (global patterns). I have already generated some examples, not always with pattern analysis.

  • Fluke switch points of higher co-dimension
    • A switch point that is simultaneously a pattern across the wage axis and a reswitching pattern (a case of a real Wicksell effect of zero).
    • A switch point combining four three-technique patterns (due to Salvadori and Steedman).
  • Fluke switch points combined with other switch points
    • A reswitching example with one switch point being a pattern across the wage axis (another case of a real Wicksell effect of zero).
    • An example with a pattern across the wage axis and a pattern over the axis for the rate of profits.
    • Two switch points with both being reswitching patterns can be found from a partition of a parameter space where two loci for reswitching patterns intersect.
    • A pattern across the point where the rate of profits is negative one hundred percent, combined with a switch point, for the same techniques, with a positive rate of profits (of interest for the reverse substitution of labor).
    • An example where every point on the frontier is a switch point.

The above list is not complete. More types of fluke switch points exist. Some, like the examples of a real Wicksell effect of zero, I thought, should be of interest for themselves to economists. Others show examples of parameters where the appearance of the wage frontier, at least, changes with perturbations of the parameters. I would like to see that in at least some cases, short run dynamics changes qualitatively with such perturbations. But this seems to be beyond my capabilities.

Maybe I'll update this post some day, if I create more examples.

Monday, January 08, 2018

A Pattern For The Reverse Substitution Of Labor

Figure 1: Variation of Switch Points with Time
1.0 Introduction

This post presents another local pattern of co-dimension one. I have conjectured that only four types of local patterns of co-dimension one exist (a reswitching pattern, a three-technique pattern, a pattern across the wage axis, and a pattern over the axis for the rate of profits). In this conjecture, I meant to implicitly limit the rate of profits at which switch points occur to be non-negative and not exceeding the maximum rate of profits. The pattern illustrated in this post is a pattern around the rate of profits of -100 percent. (I was prompted to develop this example by an anonymous comment, as I was also prompted for this post.)

Although this is a local pattern, its interest comes from global effects. Suppose another switch point exists, other than the one at a rate of profits of -100 percent. This other switch point involves the same two techniques and occurs at a positive rate of profits. A perturbation of coefficients of production around the pattern changes the other switch point from one exhibiting a conventional substitution of labor to the reverse substitution of labor. Han and Schefold (2005) describe empirical examples of the reverse substition of labor.

2.0 Technology

Table 1 specifies the technology for this example. I make the usual assumptions. Each column lists inputs per unit output for each process. Each process exhibits constant returns to scale. Each process requires a year to complete, and there are no joint products. Inputs of capital goods are totally used up in production.

Table 1: Processes of Production
InputIron IndustryCorn Industry
Labor1 Person-Yr.9/100.994653826 et
Iron(7/10) Ton1/400.002444903 et
Corn2 Bushels1/100.746512055 et

In this example, technical change occurs in the Beta process for producing corn. I assume that σ is 1/10. So coefficients of production fall at a rate of ten percent.

At any moment of time, two techniques can be created out of these processes. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Likewise, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta.

3.0 Prices and Structural Economic Dynamics

I consider a common system for defining prices of production. Relative spot prices are assumed to be constant, and the same rate of profits is earned in both industries for the cost-minimizing technique. Labor power is advanced, and wages are paid out of the surplus product at the end of the year. For a technique that is not cost-minimizing, the costs for operating the corn-producing process in this technique, as evaluated at prices of production, exceed the revenues. I take a bushel corn as the numeraire.

These assumptions allow one to construct wage curves for each technique. The cost-minimizing technique at, say, a given rate of profits is found from the outer envelopes of the wage curves, that is, the wage frontier. Switch points arise at rate of profits for which both techniques are cost-minimizing. For this example, I do not present wage curves at selected moments of time. Figure 1 graphs the rate of profits at switch points and the maximum rate of profits against time. Patterns, including a pattern for the reverse substitution of labor, are indicated on the graph.

3.1 A Superficial Neoclassical Story

Suppose one limits one's analysis to non-negative rates of profits. Figure 1 shows that technical progress leads to a switch point at the maximum rate of profits. As the wage curve for the Beta technique continues to move outward, this switch point falls below the maximum rate of profits. For rate of profits lower than at the switch point, the Alpha technique is cost-minimizing. The Beta technique is cost-minimizing at higher rates of profits. Eventually, the switch point disappears across the wage axis, and only the Beta technique is cost-minimizing.

This story initially seems to correspond to exploded neoclassical intuition about technical change. Reswitching and capital-reversing - two phenomena much emphasized in the Cambridge Capital Controversy - never occur. Around the switch point with a positive rate of profits, the Beta technique is cost-minimizing at a notionally smaller wage, and the Alpha technique is cost-minimizing at a notionally higher wage. A lower wage is associated with a technique in which greater labor inputs, aggregated across both industries, are employed per bushel of corn produced net.

3.2 A Region in which the Reverse Substitution of Labor Occurs

But consider what happens when the analysis is extended to a rate of profits of -100 percent. A switch point with a positive rate of profits exists only for time between the patterns over the axis for the rate of profits and across the wage axis. Figure 2 graphs the difference in the labor coefficients with time. After the pattern for the reverse substitution of labor, the labor coefficient for the Alpha process in producing corn exceeds the labor coefficient for the Beta process in producing corn. That is, around the switch point, the adoption of the cost-minimizing technique at a lower wage results in less labor being employed in corn production per unit corn produced gross. How is this consistent with the textbook account of labor demand functions?

Figure 2: Change in Labor Input per Unit Gross Output in Corn
4.0 Conclusion

The more I investigate price theory, the less I understand how economists can teach neoclassical microeconomics.

Friday, January 05, 2018

Labor Values As A Foundation

Figure 1: Physical Production Data as a Side Route
1.0 Introduction

One way of reading the first volume of Marx's Capital is that labor values provide a foundation, upon which the structure of prices of production and, eventually market prices are based. I find that, for example, Joseph Schumpeter presents Marx's work in this way.

Another reading takes both labor values and prices as founded on physical data specifying the technique in use. Ian Steedman, as illustrated in Figure 2, argues for such a reading. Furthermore, Steedman argues that one cannot get from the system of labor values to prices. Labor values are not needed for analyzing prices of production; they are redundant.

Figure 2: Labor Values as a Side Route

These are not the only possible ways of reading Marx. Another reading might emphasize the bits on commodity fetishism. Nothing is hidden. In selling produced commodities on the market, the concrete work activities that go into making commodities are abstracted from and treated as commensurable. This is crazy, but according to Marx, this is how capitalism works.

I seem to have stumbled on some mathematics supporting the first reading. I consider the question of what physical data is consistent with given labor values and direct labor inputs, under the condition that the organic composition of capital does not vary among industries. The issue is not that there is no way to go from labor values, through data on physical production, to prices. Rather, there are too many routes - in fact, an infinite number of them.

Figure 1 is not quite how I present my results in my draft paper. I end up with the wage curve for the price system; unlike in the above diagram, I do not close the system. I am not sure I am correct on how I specify distributive variables in the figure. I end up with the wage as a vector, where the same money wage is earned in each industry. I found it natural to close the system with the rate of profits when going from labor values to prices. On the other hand, I found it more convenient to specify the wage in going from physical data to prices. Perhaps these closures need more thought.

A substantial issue is whether it makes any sense to talk about labor values prior to and independently of physical data on processes of production. Steedman asserts it is not possible. Marx, in the first volume of Capital goes back and forth between labor values and prices. I might need to think a little more about how money, or the choice of a numeraire, fits into this, but I seem to be arguing for this possibility, at least under the conditions in which a simple labor theory of value holds as a theory of price.