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 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.
Tuesday, August 01, 2017
Switch Points Disappearing Or Appearing Over The Axis For The Rate Of Profits
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. In this case, I explore how a decrease in a coefficient of production can cause a switch point to appear or disappear over the axis for the rate of profits.
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 iron needed to produce a unit iron in the iron industry. v is the quantity of copper needed to produce a unit copper.
Input | Industry | |||
Iron | Copper | Corn | ||
Alpha | Beta | |||
Labor | 1 | 2/3 | 1 | 2/3 |
Iron | u | 0 | 1/3 | 0 |
Copper | 0 | v | 0 | 1/3 |
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.
I make all my standard assumptions. The choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Hence, wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves.
3.0 InnovationsI have two stories of technical innovation. In one, improvements are made in the process for producing copper. As a consequence, the wage curve for the Beta technique moves outward. In the other story, improvements are made in the iron industry, and the wage curve for the Alpha technique moves outwards. The bifurcations that occur in the two stories are mirror reflections of one another, in some sense.
3.1 Improvements in Copper ProductionLet u be fixed at 1/3 tons per ton. The wage curve for the Alpha technique is a downward sloping straight line. Let v decrease from 1/2 to 3/10. When v is 1/3, the wage curve for the Beta technique is also a straight line. I created the example to have linear (actually, affine) wage curves at the bifurcation for convenience. The bifurcation does not require such.
Figure 2 shows the wage curves when the copper coefficient for copper production is a high value, in the range under consideration. A single switch point exists, and the Alpha technique is cost-minimizing if the rate of profits is high. As v decreases, the switch point moves to a higher and higher rate of profits. (These statements are about the shapes of mathematical functions. They are not about historical processes set in time.) Figure 3 shows the wage curves when v is 1/3. The switch point is now on the axis for the rate of profits. For any non-negative rate of profits below the maximum, the Beta technique is cost-minimizing. Finally, Figure 4 shows the wage curves for an even lower copper coefficient in copper production. Now, there is no switch point, and the Beta technique is always cost-minimizing, for all possible prices of production.
Figure 2: Wage Curves Without Improvement in Copper Production |
Figure 3: Wage Curves For A Bifurcation |
Figure 4: Wage Curves After Improvements in Copper Production |
3.2 Improvements in Iron Production
Now let v be set at 1/3. Let u decrease from 1/2 to 3/10. Figure 5 shows the wage curves at the high end for the iron coefficient in iron production. No switch point exists, and the Beta technique is always cost-minimizing. I thought about repeating Figure 3, for v decreased to 1/3. The same configuration of wage curves, with a bifurcation, appears in this story. Figure 6, shows that the switch point appears for an even lower value of the iron coefficient.
Figure 5: Wage Curves Without Improvement in Iron Production |
Figure 6: Wage Curves After Improvements in Iron Production |
3.3 Improvements in Both Iron and Copper Industries
I might as well graph (Figure 7) the copper coefficient in copper production against the iron coefficient in iron production. The bifurcation occurs when the maximum rates of profits are identical in the Alpha and Beta technique. In a model with the simple structure of the example, this occurs when u = v. Representative illustrations of wage curves are shown in the regions in the parameter space. A switch point below the maximum rate of profits exists only above the line in parameter space representing the bifurcation.
Figure 7: Bifurcation Diagram for Two Coefficients of Production |
The story in Section 3.1 corresponds to moving downwards on a vertical line in Figure 7. The left-hand side of Figure 1, at the top of this post, is another way of illustrating this story. On the other hand, Section 3.2 tells a story of moving leftwards on a horizontal line in Figure 7. The right-hand side of Figure 1 illustrates this story.
Focus on the intersections, in the two sides of Figure 1 of the blue, red, and purple loci. Can you see that, in some sense, they are reflections, up to a topological equivalence?
4.0 DiscussionI have a reswitching example with a switch point disappearing over the axis for the rate of profits. In that example, the disappearing switch point is 'perverse', that is, it has a positive real Wicksell effect. In the examples in Section 3 above, the disappearing or appearing switch point is 'normal', with a negative real Wicksell effect.
Friday, July 28, 2017
Bifurcations Along Wage Frontier, Reflected
Figure 1: Bifurcation Diagram |
This post continues a series investigating structural economic dynamics. I think most of those who understand prices of production - say, after working through Kurz and Salvadori (1995) - understand that technical innovation can change the appearance of the wage frontier. (The wage frontier is also called the wage-rate of profits frontier and the factor-price frontier.) Changes in coefficients of production can create or destroy a reswitching example. But, as far as I know, nobody has systematically explored how this happens in theory.
I claim that when switch points appear on or disappear off of the wage frontier, these bifurcations follow a few normal forms. I have been describing each normal form as a story of a coefficient of production being reduced by technical innovation. I further claim that, in some sense, the order of changes along the wage frontier is not specified. One can find an example with a decreasing coefficient of production in which the order is the opposite of some other example of technical innovation with the same normal form.
This post is one of a series providing the proof that order does not matter. The example in this post relates to this previous example.
2.0 TechnologyThe example in this post is one of an economy in which four commodities can be produced. These commodities are called iron, steel, copper, and corn. The managers of firms know (Table 1) of one process for producing each of the first three commodities. These processes exhibit Constant Returns to Scale. Each column specifies the inputs required to produce a unit output for the corresponding industry. Variations in the parameter v can result in different switch points appearing on the frontier. Each process requires a year to complete and uses up all of its inputs in that year. There is no fixed capital.
Input | Industry | ||
Iron | Steel | Copper | |
Labor | 1/2 | 1 | 3/2 |
Iron | 53/180 | 0 | 0 |
Steel | 0 | v | 0 |
Copper | 0 | 0 | 1/5 |
Corn | 0 | 0 | 0 |
Three processes are known for producing corn (Table 2). As usual, these processes exhibit CRS. And each column specifies inputs needed to produce a unit of corn with that process.
Input | Process | ||
Alpha | Beta | Gamma | |
Labor | 1/2 | 1 | 3/2 |
Iron | 1/3 | 0 | 0 |
Steel | 0 | 1/4 | 0 |
Copper | 0 | 0 | 1/5 |
Corn | 0 | 0 | 0 |
Four techniques are available for producing a net output of a unit of corn. The Alpha technique consists of the iron-producing process and the Alpha corn-producing process. The Beta technique consists of the steel-producing process and the Beta process. The Gamma technique consists of the remaining two processes.
As usual, the choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves.
3.0 Technical ProgressFigures 2 through 5 illustrate wage curves for different levels of the coefficient of production denoted v in the table. Figure 2 shows that for a relatively high parameter value, the switch point between the Alpha and Gamma techniques is the only switch point on the outer frontier. For continuously lower parameter values of v, the wage curve moves outward. Figure 3 illustration the bifurcation value, a fluke case in which the wage curves for all three techniques intersect in a single switch point. Other than at the switch point, the wage curve for the Beta technique is not on the frontier. But, for a slightly lower parameter value (Figure 4), the wage curve for the Beta technique, along with switch points between the Alpha and the Beta techniques and between the Beta and Gamma techniques, is on the frontier. The intersection between the wage curves for the Alpha and Gamma techniques is no longer on the frontier. Figure 5 illustrates another bifurcation in the example. The focus of this post is not on this bifurcation, in which a switch point disappears over the axis for the rate of profits.
Figure 2: Wage Curves with High Steel Inputs in Steel Production |
Figure 3: Wage Curves with Medium Steel Inputs in Steel Production |
Figure 4: Wage Curves with Low Steel Inputs in Steel Production |
Figure 5: Wage Curves with Lowest Steel Inputs in Steel Production |
4.0 Discussion
The bifurcation on the right in Figure 1, at the top of this post, is topologically equivalent to the horizontal reflection of the bifurcation on the right in the equivalent figure in this previous post. (On the other hand, the bifurcations on the upper left in both diagrams are the same normal form, in the same order.)
The bifurcation described in this post is a local bifurcation. To characterize this bifurcation, one need only look at small range of rates of profits and coefficients of production around a critical value. Accordingly, then wage curves involved in the bifurcations could intercept any number of times, in some other example of this normal form, at positive rates of profits. Each of the three switch points involved in the bifurcation could have any direction for real Wicksell effects, positive or negative.
The bifurcation, as depicted in this post, replaces one switch point on the wage curve with two switch points. It could be that the switch point disappearing exhibits capital-reversing, and both of the two new switch points appearing also exhibit capital-reversing. But any of five other other combinations are possible.