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.
Tuesday, August 22, 2017
The Concept Of Totality
This post is inspired by current events
"It is not the primacy of economic motives in historical explanation that constitutes the decisive difference between Marxism and bourgeois thought, but the point of view of totality. The category of totality, the all-pervasive supremacy of the whole over the parts is the essence of the method which Marx took over from Hegel and brilliantly transformed into the foundations of a wholly new science. The capitalist separation of the producer from the total process of production, the division of the process of labour into parts at the cost of the individual humanity of the worker, the atomisation of society into individuals who simply go on producing without rhyme or reason, must all have a profound influence on the thought, the science and the philosophy of capitalism. Proletarian science is revolutionary not just by virtue of its revolutionary ideas which it opposes to bourgeois society, but above all because of its method. The primacy of the category of totality is the bearer of the principle of revolution in science.
The revolutionary nature of Hegelian dialectics had often been recognised as such before Marx, notwithstanding Hegel's own conservative applications of the method. But no one had converted this knowledge into a science of revolution. It was Marx who transformed the Hegelian method into what Herzen described as the 'algebra of revolution'. It was not enough, however, to give it a materialist twist. The revolutionary principle inherent in Hegel's dialectic was able to come to the surface less because of that than because of the validity of the method itself, viz. the concept of totality, the subordination of every part to the whole unity of history and thought. In Marx the dialectical method aims at understanding society as a whole. Bourgeois thought concerns itself with objects the arise either from the process of studying phenomena in isolation, or from the division of labour and specialisation in the different disciplines. It holds abstractions to 'real' if it is naively realistic, and 'autonomous' if it is critical."
-- Georg Lukács, History and Class Consciousness (trans. by Rodney Livingstone), MIT Press (1971): pp. 27-28.
Sunday, August 20, 2017
A Reswitching Bifurcation, Reflected
Figure 1: Two Bifurcation Diagrams Horizontally Reflecting |
This post continues my investigation of structural economic dynamics. I am interested in how technological progress can change the analysis of the choice of technique. I have four normal forms for how switch points can appear on or disappear from the wage frontier, as a result of changes in coefficients of production. This post concentrates on what I call a reswitching bifurcation.
Each bifurcation can be described by how wages curves look around the bifurcation before, at, and after the bifurcation. I claim that, in some sense, order does not matter. For each normal form, bifurcations can exist in either order. I have proven this, for three of the bifurcations, by constructing the normal forms in both orders. This post completes the proof by exhibiting both orders for the reswitching bifurcation.
2.0 TechnologyConsider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. Technology is defined in terms of two parameters, u and v. u denotes the quantity of labor needed to produce a unit iron in the iron industry. v is the quantity of labor needed to produce a unit copper.
Input | Industry | |||
Iron | Copper | Corn | ||
Alpha | Beta | |||
Labor | u | v | 1 | 361/91 |
Iron | 1/2 | 0 | 3 | 0 |
Copper | 0 | 48/91 | 0 | 1 |
Corn | 0 | 0 | 0 | 0 |
As usual, this technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta.
3.0 Selected Configurations of Wage Curves3.1 A Reswitching Bifurcation
Consider certain specified parameter values for the coefficients of production denoting the amount of labor needed to produce one unit of iron and one unit of copper. In particular, let u be 1, and let v be 17,328/8,281. Figure 2 graphs the wage curves for the two techniques in this case.
Figure 2: Wage Curves at the Bifurcation |
I call this case a reswitching bifurcation. Like all bifurcations, it is a fluke case.
3.2 Improvements in Iron Production Around The Reswitching BifurcationConsider variations in u, from some parameter larger than its value in the above reswitching bifurcation to some lower value. In this part of the story, the value of v is assumed to be fixed at its value for the bifurcation. The right half of Figure 1, at the top of this post, illustrates this story.
For a high value of u, to the right of the right of Figure 1, the wage curve for Alpha is moved inside its location in Figure 2. The wage curves for the Alpha and Beta techniques intersect at two points. It is a reswitching example. A fall in u is illustrated by a movement to the left on the right side of Figure 1. The two switch points become closer and closer along the wage frontier. The reswitching bifurcation is illustrated by the thin vertical line in Figure 1. For any smaller value of u, the Alpha technique is cost minimizing for all feasible rates of profits or wages.
3.3 Improvements in Copper Production Around The Reswitching BifurcationNow consider variations in v, with u fixed at the value for the bifurcation illustrated in Figure 2. Technical progress in the copper industry is illustrated by a movement to the left on the left side of Figure 1. For a high value of v, the wage curve for the Beta technique is inside the wage curve for the Alpha technique. The Alpha technique is cost-minimizing for all feasible rates of profits. As v decreases, the wage curve for the Beta technique moves outward, until it reaches the reswitching bifurcation. For smaller values of v, the example becomes, once again, a reswitching example. A second bifurcation is illustrated on the left side of Figure 1, when the switch point at the higher rate of profits moves across the axis for the wage. The labor input for copper has become so small that the Beta technique is cost-minimizing for any sufficiently large enough wage and small rate of profits.
4.0 ConclusionThe bifurcation depends on a certain relative configuration of wage curves, in which one is tangent to the other at a switch point. Whether technical progress around the bifurcation results in reswitching appearing or disappearing depends on which wage curve is moving outwards faster around the switch point(s). Either order is possible.
Tuesday, August 15, 2017
Elsewhere
- Nick Hanauer argues for some policies that postulate:
- Income distribution is not a matter of supply and demand or any other sort of economic natural laws.
- That a more egalitarian distribution of income leads to an increased demand and generalized shared prosperity.
- Tom Palley contrasts neoliberalism with an economic theory with an approach with another "theory of income distribution and its theory of aggregate employment determination".
- Elizabeth Bruenig contrasts liberalism with the the left.
- Paul Blest laughs at whining neoliberals
- Chris Lehmann considers how the turn of the US's Democratic Party to neoliberalism lowers its electoral prospects.
Is the distinction between democratic socialism and social democracy of no practical importance at the moment in any nation's politics? I think of the difference in two ways. First, in the United States in the 1970s, leftists had an argument. Self-defined social democrats became Neoconservatives, while democratic socialists found the Democratic Socialists of America (DSA). Second, both are reformists approaches to capitalism, advocating tweaks to, as Karl Popper argued for, prevent unnecessary pain. But social democrats have no ultimate goal of replacing capitalism, while democratic socialists want to end up with a transformed system.
Saturday, August 12, 2017
A Fluke Of A Fluke Switch Point
Figure 1: Wage Curves |
This post presents an example of the analysis of the choice of technique in competitive markets. The example is one with three techniques and two switch points. The wage curves for the Alpha and Beta techniques are tangent at one of the switch points. This is a fluke. And the wage curves for all three techniques all pass through that same switch point. This, too, is a fluke.
I suppose that the example is one of reswitching and capital-reversing is the least interesting property of the example. Paul Samuelson was simply wrong in labeling such phenomena as perverse. A non-generic bifurcation, like the illustrated one, falls out of a comprehensive analysis of possible configurations of wage curves.
2.0 TechnologyThe technology in the example has a particularly simple structure. Firms can produce one of three capital goods, which I am arbitrarily labeling iron, copper, and uranium. Table 1 shows the production processes known for producing each metal. One process is known for producing each, and each metal is produced out of inputs of labor and that metal. Each process requires a year to complete, uses up all its material inputs, and exhibits Constant Returns to Scale.
Input | Industry | ||
Iron | Copper | Uranium | |
Labor | 1 | 17,328/8,281 | 1 |
Iron | 1/2 | 0 | 0 |
Copper | 0 | 48/91 | 0 |
Uranium | 0 | 0 | 0.53939 |
Corn | 0 | 0 | 0 |
Three processes are known for producing corn (Table 2), which is the consumption good. This economy can be sustained by adopting one of three techniques. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta. Finally, the Gamma technique consists of the remaining two processes.
Input | Process | ||
Alpha | Beta | Gamma | |
Labor | 1 | 361/91 | 3.63505 |
Iron | 3 | 0 | 0 |
Copper | 0 | 1 | 0 |
Uranium | 0 | 0 | 1.95561 |
Corn | 0 | 0 | 0 |
The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage-rate of profits for the three techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. Aside from switch points, the Alpha technique is cost-minimizing at low and high rates of profits, with the Gamma technique cost-minimizing between the switch points. At switch points, any linear combination of the techniques with wage curves going through that switch point are cost-minimizing.
The wage curve for the Beta technique is a straight line. This affine property results from the Organic Composition of Capital being the same in copper production and in corn production, when the Beta technique is adopted. To help visualization, I also graph the difference between the wage curves (Figure 2). The Beta technique is only cost-minimizing at the switch point at the higher rate of profits. The tangency of the wage curves for the Alpha and Beta techniques is manifested in Figure 2 by the non-negativity of the difference in these curves.
Figure 2: Distance Between Wage Curves |
4. Conclusion
I'm sort of proud of this example. I suppose I could, at least, submit it for publication somewhere. But it is only a side effect of a larger project I guess I am pursuing.
I want to introduce a distinction among fluke switch points. Every bifurcation (that is, a change in the sequence of switch points and cost-minimizing techniques along the wage frontier) is a fluke. Some perturbation of a coefficient of production from a bifurcation value will change that sequence. Suppose a perturbation of a coefficient of production not involved in a bifurcation, in some sense, leaves the qualitative story unchanged. One can use the same bifurcation to tell a story about, say, technological progress. This is a generic bifurcation.
Accept, for the sake of argument, that prices of production tell us something about actual prices. The economy is never in an equilibrium, but owners of firms are always interested in increasing their profits. One can never expect observed technology to meet the fluke conditions of a generic bifurcation. But it can tell us something about how the dynamics of income distribution, for example, vary with technological progress.
Suppose one perturbs, in the example, the coefficient of production for the amount of iron needed to produce iron. (I denote this coefficient, in a fairly standard notation, as a_{1,1}^{β}.) Then, either the wage curves for the. Alpha and Beta techniques will not intersect at all or they will intersect twice. In the latter case, one can vary a_{1,1}^{γ} to find an example in which all three wage curves intersect at one or another of the switch points. But the tangency will be lost. So I consider the fluke point illustrated to be a non-generic bifurcation.
Non-generic bifurcations arise in a complete bifurcation analysis. The model illustrated remains open. Income distribution is not specified. Nevertheless, I think this theoretical analysis can say something to those who are attempting to empirically apply the Leontief-Sraffa model.
Monday, August 07, 2017
Some Unresolved Issues In Multiple Interest Rate Analysis
Come October, as I understand it, the Review of Political Economy will publish, in hardcopy, my article The Choice of Technique with Multiple and Complex Interest Rates. I discuss in this post questions I do not understand.
2.0 Non-Standard Investments and Fixed CapitalConsider a point-input, flow-output model. In the first year, unassisted labor produces a long-lived machine. In successive years, labor and a machine of a specific history are used to produce outputs of a consumption good and a one-year older machine. The efficiency of the machine may vary over the course of its physical lifetime. When the machine should be junked is a choice variable in some economic models.
I am aware that in this, or closely related models, the price of a machine of a specific date can be negative. The total value of outputs at such a year in which the price of the machine of the machine is negative, however, is the difference between the sum of the price of the machine & the consumption good and the price of any inputs, like labor, that are hired in that year. Can one create a numerical example of such a case in which the net value, in a given year, is negative, where that negative value is preceded and followed by years with a positive net value?
If so, this would an example of a non-standard investment. A standard investment is one in which all negative cash flows precede all positive cash flows. In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments create the possibility that all roots of the polynomial used to define the Internal Rate of Return (IRR) are complex. Can one create an example with fixed capital or, more generally, joint production in which this possibility arises?
Does corporate finance theory reach the same conclusions about the economic life of a machine as Sraffian analysis in such a case? Can one express Net Present Value (NPV) as a function combining the difference between the interest rate and each IRR in this case, even though all IRRs are complex? (I call such a function an Osborne expression for the NPV.)
3.0 Generalizing the Composite Interest Rate to the Production of Commodities by Means of CommoditiesIn my article, I follow Michael Osborne in deriving what he calls a composite interest rate, that combines all roots of the polynomial defining the IRR. I disagree with him, in that I do not think this composite interest rate is useful in analyzing the choice of technique. But we both obtain, in a flow-input, point output model, an equation I find interesting.
This equation states that the difference between the labor commanded by a commodity and the labor embodied in that commodity is the product of the first input of labor per unit output and the composite interest rate. Can you give an intuitive, theoretical explanation of this result? (I am aware that Osborne and Davidson give an explanation, that I can sort of understand when concentrating, in terms of the Austrian average period of production.)
A model of the production of commodities by means of commodities can be approximated by a model of a finite sequence of labor inputs. The model becomes exact as the number of dated labor inputs increases without bound. In the limit, the labor command by each commodity is a finite value. So is the labor embodied. And the quantity for the first labor input decrease to zero. Thus, the composite interest rate increases without bound. How, then, can the concept of the composite interest rate be extended to a model of the production of commodities by means of commodities?
4.0 Further Comments on Multiple Interest Rates with the Production of Commodities by Means of CommoditiesIn models of the production of commodities by means of commodities, various polynomials arise in which one root is the rate of profits. I have considered, for example, the characteristic equation for a certain matrix related to real wages, labor inputs, and the Leontief input-output matrix associated with a technique of production. Are all roots of such polynomials useful for some analysis? How so?
Luigi Pasinetti, in the context of a theory of
Structural Economic Dynamics,
has described what he calls the natural system.
In the price system associated with the natural system, multiple interest rates arise, one for each produced commodity.
Can these multiple interest rates be connected to Osborne's natural multiple interest rates?
I would not mind reading attempts to answer the above questions.
Friday, August 04, 2017
Switch Points and Normal Forms for Bifurcations
I have put up a working paper, with the post title, on my Social Sciences Research Network (SSRN) site.
Abstract: The choice of technique can be analyzed, in a circulating capital model of prices of production, by constructing the wage frontier. Switch points arise when more than one technique is cost-minimizing for a specified rate of profits. This article defines four normal forms for structural bifurcations, in which the number and sequence of switch points varies with a variation in one model parameter, such as a coefficient of production. The 'perversity' of switch points that appear on and disappear from the wage frontier is analyzed. The conjecture is made that no other normal forms exist of codimension one.