Wednesday, January 01, 2020
Welcome
The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.
In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.
I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.
Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.
Saturday, January 20, 2018
Labor Values Taken As Given
I have been considering a case in which a simple Labor Theory of Value (LTV) is a valid theory of prices of production. When, for each technique, all processes have the same organic composition of capital, prices of production are proportional to labor values. Given labor values and direct labor coefficients in each industry, an uncountably infinite number of techniques - as specified by a Leontief input-output specified in terms of physical inputs per physical outputs - satisfies these conditions.
In outlining this mathematics, I start with labor values and derive technical conditions of production as a detour on the way to prices of production. (I have also considered a perturbation of this possibility, as an application of my pattern analysis.)
Has anybody commenting on Marx actually started with labor values, taken as given, in this way? If this is a straw person, I am good company. Ian Steedman (1977) makes something like the same accusation. See the section, "A spurious impression", in Chapter 4, "Value, Price, and Profit Further Considered", of his book.
But I have found examples of other approaching Marx in something like this way. I refer to von Bortkiewicz (1907) and Seton (1957), two authors taken as a precursor to the Sraffian reading of Marx. The fact that Steedman can be read as criticizing such authors complicates the claim that this literature exhibits continuity. I think others have also argued that some novelty arises in Steedman's critique insofar as he argues that labor values are redundant, since prices of production are properly calculated from technical data on production and the physical composition of wage goods.
Perhaps my examples of Bortkiewicz and Seton should not be read as propounding any large claim that Marx takes labor values as more fundamental, in some sense, than physical conditions in production processes. Rather, Bortkiewicz started from the schemes of simple and expanded reproduction at the end of Volume 2 of Capital. Since Seton, and other authors, were generalizing and commenting on Bortkiewicz, they, as a matter of path dependence, happened to keep the assumption of given labor values. One wanting to argue for a reading of Marx that I seem to be stumbling into, without any firm commitment, needs to deal with Volume 1.
I have two additional notes on rereading these references. First, I like to talk about Marx' invariants in the transformation problem. I thought I had taken this term from formal modeling in computer science. Edsger Dijkstra and C. A. R. Hoare talk about loop invariants, and I sometimes even comment my code with explicit statements of invariants. But Seton has a section titled "Postulates of Invariance".
Is Steedman disappointed in the reception of his book? Obviously, his points about the transformation problem, including the possibility of negative surplus value being consist with positive profits, under a case of joint production, have been widely discussed. But consider his exposition of simple examples intended to demonstrate that Sraffa's analysis can take into account all sorts of issues that some had argued were ignored. Consider letting how much work capitalists can get out of labor being a variable, heterogeneous types of abstract labor not reducible to one and the possibility of workers of each type exploiting others, wages being paid, say, weekly, during processes that take a year to complete, how wages relate to the rate of exploitation when a choice of technique exists, the treatment depreciation of capital, and the existence of a retail sector for circulating produced commodities. How many of these analyses have been taken up and continued by those building on Sraffa? (I think some have.)
References- Eugen von Bohm-Bawerk (1949). Karl Marx and the Close of his System: Bohm-Bawerk's Criticism of Marx. Edited by P. M. Sweezy.
- Ladislaus von Bortkiewicz (1907). On the Correction of Marx's Fundamental Theoretical Construction in Third Volume of Capital, Trans. by P. M. Sweezy. In Bohm-Bawer (1949).
- F. Seton (1957). The "Transformation" Problem. Review of Economic Studies, 24 (3): 149-160.
- Ian Steedman (1981, first edition 1977). Marx After Sraffa, Verso.
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 |
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 TechnologyTable 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.
Input | Iron Industry | Corn Industry | |
Alpha | Beta | ||
Labor | 1 Person-Yr. | 9/10 | 0.994653826 e^{-σ t} |
Iron | (7/10) Ton | 1/40 | 0.002444903 e^{-σ t} |
Corn | 2 Bushels | 1/10 | 0.746512055 e^{-σ t} |
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 DynamicsI 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 StorySuppose 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 OccursBut 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 |
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 |
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.
Sunday, December 31, 2017
Perturbation Of An Example With A Continuum Of Switch Points
Figure 1: A Partitioning Of The Parameter Space |
I consider here a case where two different techniques have the same wage curve. A simple labor of theory of value describes prices in the case under consideration. I treat the labor coefficient and another coefficient of production for a process in one technique as parameters. And I look at what happens when they vary.
A note on terminology: on the basis of expert advice and peer review, I am no longer using the term "bifurcation" for a pattern of switch points where a perturbation of model parameters, such as coefficients of production, removes or adds a switch point to the wage frontier. Instead, I am calling such a configuration a "pattern."
2.0 TechnologyTable 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.
Input | Iron Industry | Corn Industry | |
Alpha | Beta | ||
Labor | (1/8) Person-Yr. | u | (1/2) |
Iron | (1/2) Ton | v | 2 |
Corn | (1/16) Bushel | (1/80) | (1/4) |
Two techniques can be created out of these processes. The Alpha technique consists of the iron-producing process labeled Alpha and the corn-producing process. Likewise, the Beta technique consists of the iron-producing process labeled Beta and the corn-producing process.
I think I'll say something about how I created this example. A simple labor theory of value applies to the Alpha technique. The wage curves associated with the Alpha and Beta techniques are identical when u = (1/8) person-year and v = (7/10) ton. This special case is an application of some math I have set out in a working paper.
3.0 PricesI take corn as the numeraire and assume labor is advanced. Wages are paid out of the surplus at the end of year.
3.1 AlphaThe price of production for iron, when the Alpha technique is in use, is:
p_{α} = (1/4)
The wage curve for the Alpha technique is:
w_{α} = (1/2)(1 - 3 r)3.2 Beta
The price of production for iron, when the Beta technique is in use, is:
p_{β} = [(1 + 120u) + (1 - 40u)r)]/{80[(4u - v)r + (1 + 4u - v)]}
The wage curve for the Beta technique is:
w_{β} = [(10v - 1)r^{2} -4(5v + 3)r + (29 - 30v)]3.3 Switch Points/{20[(4u - v)r + (1 + 4u - v)]}
One finds switch points by equating the two wage curves:
w_{α} = w_{β}
One obtains a quadratic equation in the rate of profits, r:
+ (120 u - 20 v - 1) (r_{switch})^{2}+ (80 u - 40 v +18) r_{switch}+ (-40 u - 20 v + 19) = 0
This equation can be factored:
(r + 1)[(120 u - 20 v - 1)r + (-40 u - 20 v + 19)] = 04.0 Special Cases
4.1 A Continuum of Switch Points
I first want to check the special case u = (1/8) and v = (7/10). Recall, this example was created so that the wage curves for the two techniques would be identical in this case. In this special case, the two coefficients for the second factor of the Left Hand Side of the above quadratic equation reduce to zero. So that equation is identically true for all feasible rates of profits. Every point on the wage frontier is a switch point.
4.2 A Pattern Over the Wage AxisIn this pattern, a switch point exists on the wage frontier for a rate of profits of zero. That is, one must be able to factor out r from the left-hand side of the above quadratic equation. In other words, the constant term for the second factor must be zero. One thereby obtains:
-40 u - 20 v + 19 = 0
Or
v = - 2 u + (19/20)4.3 A Pattern Over the Axis for the Rate of Profits
In this pattern, the wage curves have a switch point at the maximum rate of profits. I did not start with the quadratic equation for this special case. The maximum rate of profits for the Alpha and Beta techniques are equal when the two wage curves are identical. The maximum rate of profits for the Beta technique does not depend on the value of labor coefficients. Thus, the condition for this pattern is:
v = (7/10)
You can check that above condition yields a switch point at a rate of profits of (1/3).
4.4 A Reswitching PatternIn a reswitching pattern, the wage curves for two techniques are tangent at a switch point. For the example in which only two commodities are produced, the quadratic equation obtain by equating the wage curves for two techniques has two repeated roots. In other words, the discriminant for this quadratic equation must be equal to zero. Some algebra gives (some Octave code was useful here):
400 (8 u - 1)^{2} = 0
Or:
u = (1/8)
If v is not equal to (7/10), the above value of u results at repeated roots for a rate of profits of -1. But I am only considering non-negative rates of profits. Thus, a reswitching pattern does not exist for this example at feasible rates of profits.
5.0 VisualizationI can bring the above observations together with various pictures here.
5.1 Variation of Switch Points with Coefficients of ProductionConsider how the wage frontier varies with v, given a particular parameter value of u. (After reading this section, one might consider how the wage frontier varies with u, given a particular value of v.)
Suppose u is smaller than the special case in which the wage curves for the two techniques are identical for a specific value of v. Figure 2 illustrates this case. In a certain region of variation in v, the wage curves for the Alpha and Beta techniques appear on the frontier, with a single switch point between them.
Figure 2: Variation of v, Case 1 |
As u increases, towards 1/8, the interval for v in which both wage curves appear on the frontier gets smaller and smaller. Figure 3 shows that, in the limit, this interval narrows to a width of zero. Both wage curves are identical. The wage frontier consists of a continuum of switch points.
Figure 3: Variation of v, Case 2 |
As u increases beyond 1/8, an interval for v once again appears in which the wage frontier contains a single switch point. As shown in Figure 4, the the endpoints of the interval have become interchanged, in some sense.
Figure 4: Variation of v, Case 3 |
Figure 1, at the top of this post, graphs the parameter space for u and v. The patterns across the wage axis and the over the axis for the rate of profits divide the parameter space into the four numbered regions. Table 2 lists the switch points and the techniques along the wage frontier, in each region, in order of an increasing rate of profits.
Region | Switch Points | Techniques |
1 | None | Alpha |
2 | One | Beta, Alpha |
3 | None | Beta |
4 | One | Alpha, Beta |
My methods for pattern analysis and visualization apply in this case generalizing an instance in which two techniques have identical wage curves.
6.0 ConclusionsI have conjectured that four types of patterns of co-dimension one exist (the three-technique pattern, the pattern over the wage axis, the pattern over the axis for the rate of profits, and the reswitching pattern). This example of two techniques having identical wage curves is not a counter-example to this conjecture. It is simultaneously a a pattern over the wage curve and a pattern over the axis for the rate of profits. Thus, it is at least of co-dimension two.
The conditions for those two patterns, however, are not sufficient for this pattern. They are merely necessary. One could have two wage curves with switch points on the wage axis and on the axis for the rate of profits, but differing for all positive rate of profits less than the maximum rate of profits. The example does make me wonder about my distinction between local and global patterns; this is not the type of global pattern I had in mind when I came up with the idea. And what is the co-dimension for this pattern? Is it of an uncountably infinite co-dimension?
I can see why some might think my write-up is not all that exciting. Likewise, there is a certain amount of tedium in performing the analysis documented above. Nevertheless, I was intrigued to find the above picture emerging. I think I have stumbled upon a vast unexplored landscape in which complicated fluke cases can fit.
Wednesday, December 27, 2017
Elsewhere
- Steve Keen and others, in a showy bit of performance art in London, have called for a reformation of economics. Imitating Luther, they have nailed some theses to a door. Here's some links:
- Guardian article
- Letters to the editor in response.
- New Weather Institute blog post.
- 33 Theses.
- An article by Ben Chu, in the Independent saying, more or less, let's not get carried away.
- I do not know who Charles Mudede is or what his platform is. His style is more popular and very different from mine. Examples:
- On Seattle's minimum wage, in which he brings up an imperfectionist thesis related to the Cambridge Capital Controversy.
- On Cornel West vs. Ta-Nehisi Coates. I think the idea that identity politics associated with post modernism accommodates neoliberalism is not new (see references below). I don't want to box Coates in, but the way he writes about the Black body in Between the World and Me is definitely a post modern trope. But he writes about it, I guess, because it make sense of his lived experience.
- I stumble upon a tweet by Duncan Weldon, in which he says he resolves every year to try and understand the Cambridge Capital Controversy.
- Samir Amin (1998). Spectres of Capitalism: A Critique of Current Intellectual Fashions, Monthly Review Press.
- Terry Eagleton (1996). The Illusions of Postmodernism, Blackwell.