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.
Wednesday, October 23, 2019
The Labor Theory of Value and Sraffa's Standard Commodity with Markup Pricing
I have uploaded a working paper with the post title.
Abstract: This article demonstrates relationships that are transparent in Sraffa's standard system hold even when relative rates of profit vary persistently among industries. Even with such variations, total constant capital, total variable capital, total surplus value, and the rate of profits are unaltered by evaluation at labor values and at prices of production in Sraffa’s standard system. These results buttress those who see in the standard commodity a solution for Marx’s so-called transformation problem.
Saturday, October 19, 2019
Actually Existing Socialism In A Capitalist Setting?
Elements of a post capitalist society are and have been developing in actually existing capitalism. This post points out a couple of examples.
The Green Bay Packers is a community-owned (non-proper) football team in the National Football League (NFL). One can find some arguing that they are socialist. And some are concerned to refute this claim.
Decades ago, some universities in the United States set up research and development organizations that then became independent, not-for-profit companies. For example, here is the web site for SRC, formerly Syracuse Research Corporation. This means, apparently, that they re-invest what they make. IRS Publication 557 explains how to apply for status as a 501(c) organization.
A quick Google search gets me to the National Center for Employee Ownership. They explain how a Employee Stock Ownership Plan (ESOP) works.
The cooperative movement is of interest in this context. I gather the Mondragon Corporation, in Spain, is the most well-known example. But I want to turn to producer cooperatives in dairy. The Lowville Producers Dairy Cooperative is one near me. Apparently, the National Milk Producers Federation is a federation of such cooperatives. The United States Department of Agriculture (USDA) provides background. I see that they confirm what I know anecdotally, that not all dairy farmers are members of a coop.
I guess some theory is needed to make sense of any claim that, say, producer coops are an example of socialism or to obtain a general understanding of such organizations. I have only read Hodgson (1998) and Jossa (2005) in the list of references below. From Hodgson, as I recall, I learned that an issue with cooperatives is start-up finance. It may be that producer cooperatives are more efficient than capitalist firms and still be smaller than one would hope. Jossa (2005) argues that cooperatives are consistent with Marx's vision. He draws on Vanek's distinction between worker-managed firms (WMFs) and labor-managed firms (LMFs). In WMFs, workers provide the finance, while in a LMF, the firm borrows. Anyways, here is some literature to explore.
References- Geoffrey M. Hodgson (1998). Economics and Utopia: Why the Learning Economy is not the End of History. Routledge.
- Bruno Jossa (2019). The Political Economy of Cooperatives and Socialism, Routledge.
- Bruno Jossa (2005). Marx, Marxism and the cooperative movement. Cambridge Journal of Economics 29: 3-18.
- Jaroslav Vanek (1970). The General Theory of Labor-Managed Market Economies. NCOL.
- Jaroslav Vanek (1971). The Participatory Economy: An Evolutionary Hypothesis and a Strategy for Development. Cornell University Press.
- Jaroslav Vanek (1977). The Labor-Managed Economy: Essays. Cornell University Press.
Thursday, October 10, 2019
Structural Economic Dynamics and Fake Switch Points
Figure 1: A Pattern Diagram with Joint Production |
This post completes an example. I analyzed bits of this example here and here. This post may make no sense if you have not read a long series of previous posts or, maybe, the papers highlighted here and here. I am interested in how and if my approach to analyzing and visualizing variations in the choice of technique with technical progress extends to joint production. The example suggests fake switch points do not pose an insurmountable obstacle for such an extension.
2.0 TechnologyI repeat the specification of technology.
I postulate an economy in which two commodities, corn and linen, can be produced from inputs of corn, linen, and labor. Managers of firms know of three processes (Tables 1 and 2) to produce corn and linen. Each process produces net outputs of corn and linen as a joint product. Inputs and outputs are specified in physical units (say, bushels and square meters) per unit level of operation of the given process. Inputs are acquired at the start of the year, and outputs are available for sale at the end of the year.
Input | Process | ||
(a) | (b) | (c) | |
Labor | e^{σ0,1(1 - t)} | e^{σ0,2(1 - t)} | e^{σ0,3(1 - t)} |
Corn | 20 | 20 | 30 |
Linen | 20 | 20 | 30 |
Output | Process | ||
(a) | (b) | (c) | |
Corn | 21 | 23 | 36 |
Linen | 27 | 25 | 34 |
I assume that requirements for use are such that two processes must be operated to satisfy those requirements. I need to investigate the implications of this assumption further. Apparently, for this example, it implies that the economy is not on a golden rule steady state growth path, with the rate of profits equal to the rate of growth. Anyway, with this assumption, three techniques - Alpha, Beta, and Gamma - can be operated. Table 3 specifies which processes are operated for each technique.
Techniques | Processes |
Alpha | a, b |
Beta | a, c |
Gamma | b, c |
The technology, as I have defined it, is parameterized. I consider the following specification for the rate of decrease in labor coefficients.
σ_{0,1} = 2
σ_{0,2} = σ_{0,3} = 5/2
Bidard & Klimovsky's example arises when t is unity.
3.0 Prices and the Choice of TechniqueA system of two price equations arises, for each technique. I assume the labor coefficient is treated as a constant over the period of production - say, a year. With linen as numeraire, these equations for the Alpha technique are:
(20 p_{1} + 20)(1 + r) + [e^{σ0,1(1 - t)}] w = 21 p_{1} + 27
(20 p_{1} + 20)(1 + r) + [e^{σ0,2(1 - t)}] w = 23 p_{1} + 25
One can these equations for two variables in terms of, say, the rate of profits. For each technique, its wage curve shows the wage as a function of the rate of profits. One cannot generally base the choice of technique, under joint production, on figuring out which technique contributes to the outer frontier at a given rate of profits.
Instead, one can calculate profits and losses, with the given rate of profits and a technique's price system for the processes not in the technique. This exercise only makes sense when the rate of profits, the wage, and prices are non-negative for the starting technique. The technique is cost-minimizing only if no extra profits can be made with processes outside the technique.
I deliberately frame this as a combinatorial argument. Bidard likes what he calls a market algorithm, where, when one identifies a process earning extra profits, one introduces the process into the technique. In the case of joint production, it is not clear which process should be dropped. Furthermore, examples exist in which a cost-minimizing technique exists but cannot be reached from certain starting points with the market algorithm.
4.0 PatternsI have constructed the figure at the top of the post to illustrate how the choice of technique varies with technical progress in this example. The dashed lines highlight features of the example that do not bear on the choice of technique. The light vertical solid lines divide time into numbered regions. Table 3 lists the cost-minimizing techniques, in order of an increasing rate of profits in each region.
Regions | Techniques |
1 | Gamma, No Production, Alpha |
2 | Gamma, No Production, Alpha |
3 | Gamma, Alpha & Gamma, Alpha |
4 | Alpha & Gamma, Alpha |
5 | Beta, Alpha & Gamma, Alpha |
I could say a lot more about the example. I will note that in region 1, the wage increases with the rate of profits, for the Alpha technique, in the interval for the rate of profits where both wages and the price of corn are positive. In region 2, the wage decreases with the rate of profits, for the Alpha technique. The division between regions 2 and 3 is associated with that interval for the rate of profits for Alpha transitioning to have a non-empty intersection with the similar interval for the Gamma technique. for
5.0 ConclusionThis post has illustrated that one type of my types of pattern diagrams can apply to joint production. This type illustrates how the relationship between the choice of technique and distribution varies with technical progress. It can be constructed even in cases, such as joint production, where the choice of technique cannot necessarily be based on wage-rate of profits curves and their outer frontier.
If fake switch points are not shown, this type of pattern diagram does not depend on the specification of the numeraire. If the ordinate in Figure 1 were the wage, instead of the rate of profits, it would be upside down, in some sense. A different numeraire would rescale the wage. When corn is numeraire, only one fake switch point exists. It, too, would be a horizontal line segment. But fake switch points are fake precisely because they do not impact the choice of technique. They can be left off the diagram.
The example also illustrates new types of patterns for dividing adjacent regions. Under joint production, a technique can be associated with non-negative prices and a wage for an interval of the rate of profits that does not include a rate of profits of zero. Both the Alpha and the Beta technique exhibit this possibility in the example. And we can divide regions based on when the range of rate of profits in which such a technique becomes cost-minimizing comes to include zero or begins to interact with the range in which another technique is cost-minimizing
This example also illustrates that the cost-minimizing technique may not be unique in a range of rates of profits. I think this non-uniqueness is qualitatively different than how non-uniqueness can arise in models with only circulating capital. In circulating capital models, non-uniqueness is associated with two techniques having identical wage curves. Not so here.
I do not intend to write this example up any more extensively. I have no so-called paradoxical behavior here, such as reswitching, reverse capital-deepening, or the reverse substitution of labor. I may go on to explore where techniques are described by rectangular matrices, with more produced commodities than processes, and there is a dependence on the requirements for use.
References- Bidard, Christian and Edith Klimovsky (2004). Switches and fake switches in methods of production. Cambridge Journal of Economics. 28 (1): 89-97.
Saturday, October 05, 2019
Elsewhere
- Here is a post from a blog devoted to cybercommunism. The blogger is glowing about Paul Cockshoot's work on refuting Hayek's supposed refutation of the possibility of a post-capitalist society.
- William Milberg writes about how it is becoming more common to use the word "capitalism", a word mainstream economists had mostly stopped using.
- Herbert Giants and Rakesh Khurana write about the corrupting effects of neoclassical economics on what is taught in business school and then practiced by corporate elites.
- Osita Nwanevu writes, in The New Republic, about the enthusiasts that showed up at last weekend's Third MMT Conference.
- Lisa Schweitzer studies urban environments. In a blog post, she expresses irritation at Paul Romer's arrogance, admittedly filtered through a glowing New York Times article.
- A long time ago, Connie Bruck profiled George Soros in the New Yorker. Soros consciously thinks of himself as building on Karl Popper's The Open Society and its Enemies.
Saturday, September 28, 2019
Variation in Standard Commodity with Relative Markups
I am not sure about the economic logic in this post. Maybe somebody like D'Agata or Zambelli could do something with this. These ideas were suggested to me by email with a sometime commentator.
I start out with notation for Sraffa's price system, modified in an unusual way to allow for persistent variations in the rate of profits among industries:
- a_{0} is a row vector of labor coefficients in each of n industries.
- A is a Leontief input-output matrix, where a_{i, j} is the quantity of the ith commodity needed as input to produce one unit of the jth commodity.
- S is a diagonal matrix, where all off-diagonal elements are zero. s_{j, j} is the markup on non-labor costs in the jth industry.
- p is a row vector of prices.
- w is the wage.
- r is the scale factor for the rate of profits.
The coefficients of production, as expressed in the labor coefficients and the Leontief matrix are given parameters. Relative markups are also taken as given. Prices, the wage, and the scale factor for the rate of profits are the unknowns to be determined. My problem is to find a numeraire such that the wage and the scale factor for the rate of profits trade off in a straight-line relationship, at least when labor is advanced and wages are paid out of the net product:
r = R (1 - w)
I assume all elements of A are non-negative and that all elements of a_{0} and all diagonal elements of S are positive. The economy is assumed to be viable, that is, as capable of producing a surplus product. For simplicity, assume that the Leontief matrix is indecomposable. More generally, I need A S to be a Sraffa matrix.
For my purposes here, I formulate price equations as so:
p A S (1 + r) + a_{0} w = p
Consider the case when wages are zero and the scale factor for the rate of profits is at its maximum R:
p A S (1 + R) = p
Or:
p A S = (1/(1 + R)) p
I observe that prices are a left-hand eigenvector of the matrix A S, with (1/(1 + R)) the corresponding eigenvalue. To ensure that prices are positive, of the n eigenvalues, choose the maximum. The maximum eigenvalue is also known as the Perron-Frobenius root of A S.
Let y^{*} be a right-hand eigenvector of A S corresponding to its Perron-Frobenius root. Let q^{*} be gross output such that the net output is y^{*}:
y^{*} = q^{*} - A q^{*}
These quantities flow define the standard system here, when scaled so as employ a unit quantity of labor:
a_{0} q^{*} = 1
The net output of the standard system is the desired numeraire:
p y^{*} = 1
With this definition of the standard system, the ratio of physical gross outputs to circulating capital inputs varies among commodities. This result contrasts with Sraffa's standard system. I suppose I could restore this property by choosing q^{*}, not y^{*}, to be an eigenvector. Either way, the ratio of net outputs to circulating capital inputs varies among industries. Either way, the relative ratios of commodities in the standard industry depends on relative markups.
Do Marx's invariants hold with the above definition of the standard system? I expect not. Nevertheless, does this mathematics provide some insight into classical or Marxist political economy?
Saturday, September 21, 2019
A Fluke Case Over The Wage Axis
Figure 1: Wage Curves and The Price of Corn for the Fluke Case |
This post extends a previous post. I am basically introducing structural dynamics into an example, by Bidard and Klimovsky of fake switch points.
At a rate of profits of zero in the example, the price of corn is zero for Alpha, one of the two techniques that is cost-minimizing there and for somewhat higher rates of profits. At a time before the fluke case, only the Gamma technique is cost-minimizing at a rate of profits of zero. The price of corn, as calculated with the Alpha technique, is negative at a rate of profits of zero. Alpha prices become non-negative only for positive rates of profits. This possibility cannot arise in examples with only single production and the choice of technique analyzed by the construction of the wage frontier.
2.0 A Fluke CaseTechnology and techniques are specified as in the previous post. I consider variations in labor coefficients with time. Two commodities can be produced jointly with each of three production process. In each process, workers produce outputs of the two commodities from smaller inputs of each commodities. Requirements of use are such that at least two processes must be operated. So each technique combines two processes.
A system of price equations is associated with each technique. The system, including an equation specifying the numeraire, can be taken to define the wage and the prices of both commodities, given an exogenous specification of the rates of profits. Table 1, at the head of this post, illustrates the solution prices at a given point of time. Linen is taken as the numeraire. The top half of the figure shows the wage, for each technique, as a function of the rate of profits. The bottom half of diagram shows the corresponding price of corn. Notice that, for the Alpha technique, the price of corn is zero when the rate of profits is zero.
A technique is only feasible, for the analysis of the choice of technique, when both the wage and prices are non-negative. In Figure 1, the rate of profits is partitioned into two roman-numbered regions. In Region II, both the Alpha and Gamma techniques are feasible. In Region III, only the Alpha technique is feasible.
At a switch point:
- The wage curves for at least two techniques intersect at the switch point.
- No extra profits can be made at the going rate of profits in any process.
- No excess costs arise for any process that can be operated at the switch point.
No switch points exist in the example at the time illustrated in Figure 1. For the structure of the example, all three wage curves intersect at a (non-fake) switch point. Furthermore, the price of corn is the same for all three techniques at the switching rate of profits.
Not enough information has been given so far to determine which techniques are cost-minimizing at each feasible rate of profits in Figure 1. I like to plot extra profits for each process and each price system. I do not show such plots in the post. Nevertheless, Table 1 summarizes which techniques are cost minimizing.
Regions | Cost-Minimizing Technique | Processes |
I | Gamma | b, c |
II | Alpha and Gamma | a, b, c |
III | Alpha | a, c |
Consider time before the fluke case illustrated in Figure 1. Labor coefficients are larger. Figure 2, below, illustrates the wage and price curves for a specified time before the fluke case described above. Notice the appearance of Region I, where Gamma is uniquely cost-minimizing. The fluke case is a knife-edge case where Region disappears. The wage axis becomes the boundary between Regions I and II. Of these two regions, only Region II exists for a positive rate of profits.
Figure 2: A Fluke Fake Switch Point? |
Does Figure 2 illustrate another fluke case? At the fake switch point at a rate of profits of five percent, the price of corn is zero. But consider Figure 3 below. The only difference in the example between Figures 2 and 3 is the specification of the numeraire. With corn as numeraire, the fake switch points disappear, and a new fake switch point appears at a rate of profits of 13 1/3 percent. The wage and the price of linen approach an asymptote for the rate of profits at which the price of corn is zero when linen is the numeraire. Which techniques are cost-minimizing is unaffected by the choice of the numeraire.
Figure 3: Not A Fluke With Corn As Numeraire |
Consider some time after the fluke case illustrated in Figure 1. With the chosen parameters, labor coefficients have decreased less in the first production process than in the other two. Figure 4 shows the next qualitative change in the example, in which a switch point appears over the wage axis. I have already analyzed this case for this example.
Figure 4: A Switch Point On The Wage Axis |
Between the times illustrated by Figures 1 and 4, Regions II and III continue to characterize the range of feasible non-negative rates of profits. The price of corn is positive, for all three techniques, is positive for feasible rates of profits for each technique. Region I has vanished.
The switch point continues to exist after the time illustrated in Figure 4, but at a positive rate of profits. A new region appears. For a rate of profits of zero and small positive rates of profits, the Beta technique is uniquely cost-minimizing.
5.0 ConclusionThis post has presented a fluke case only possible under joint production. In this example, the choice of technique cannot be determined by constructing the wage frontier.
This post has also presented a sort-of fluke case associated with a fake switch point. In this case, the fake switch point appears on the frontier at a rate of profits at which the price of corn is zero. The set of cost-minimizing techniques and processes varies at the fake switch point. But its existence depends on the choice of the numeraire.
I have been working on a taxonomy of fluke switch points for understanding structural economic dynamics. This post illustrates that my approach can extend to joint production. New phenomena and fluke cases can arise, and one must, perhaps, pay closer attention to what is and is not dependent on the choice of the numeraire.