Wednesday, January 01, 2020


I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

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.

Monday, February 19, 2018

One Technique Replacing Another

Figure 1: One Way One Technique Can Replace Another

The wage-rate of profits frontier (or wage frontier) is calculated with prices of production, given the techniques of production, available in the economy, for producing a given output. Suppose at one point in time, the techniques that lie along the wage frontier consist of the Alpha, Beta, and Gamma techniques, in order of an increasing rate of profits. As time passes, technical innovation alters coefficients of production, including for techniques that were not on the wage frontier at the initial point in time. Suppose at a later point in time, the techniques along the wage frontier now consist of the Alpha, Delta, and Gamma techniques. How did this replacement of the Beta technique by the Delta technique occur? What happened in the intervening time interval?

The pattern analysis I have been developing suggests answers to these questions. A pattern is a qualitative characterization of a part of the wage frontier associated with a change of switch points. And the patterns I have identified suggest three possibilities for the postulated change in techniques.

Figure 1, above, illustrates the first possibility. These illustrations are only schematic; the illustrated curves need not be straight lines. In the first pattern, two three-technique patterns succeed one another in time. A three-technique pattern arises when a switch point on the frontier is an intersection of the wage curves for three techniques. For the temporally first three-technique pattern, a switch point is replaced by two switch points, with a new technique being cost-minimizing for rates of profits between the two switch points. Four techniques, instead of three techniques, now lie on the wage frontier. For the later three-technique pattern, two switch points are replaced by one switch point. The middle technique at the original point in time is no longer cost-minimizing, for any rate of profits. The postulated initial sequence of techniques occurs before the first pattern, and the final sequence occurs after the second pattern.

A second possibility is that a reswitching pattern is followed by two three-technique patterns. (The shape of the curves are definitely off in Figure 2.) For the reswitching pattern, a new switch point occurs at which the wage curves for two techniques are tangent. For some time afterwards, this possibility is a case of reswitching. The two three-technique patterns remove the wage curve for the originally middle technique from the wage frontier. Once again, the postulated observations for the first and last point in time are consistent with this story.

Figure 2: A Second Way One Technique Can Replace Another

A third possibility (Figure 3) also involves a sequence of two three-technique patterns. In this case, the temporally first three-technique pattern removes the wage-curve for the middle technique from the wage frontier. The wage frontier now has a succession of two cost-minimizing techniques along it. The second three-technique pattern puts a new technique in the middle of the wage frontier.

Figure 3: A Third Way One Technique Can Replace Another

Two other stories can arise out of symmetries of, at least, the first and second possibility. And one might complicate the story by superimposing wage curves for other techniques somewhere in this story. In my exploration of numerical examples, I have usually found the switch points in reswitching examples disappearing with patterns over the axis for the rate of profits and the wage axis. That is not the case here.

I think that wage curves can be calculated from Leontief matrices, as derived from the National Income and Product Accounts (NIPA). Zonghie Han & Bertram Schefold and Stefano Zambelli have calculated wage frontiers from empirical data. But I think confirming these stories of technical change fit data is a challenge for those who know more about empirical research following on from Leontief's work. I suspect one would have to look at stylized facts, in some sense. I think I have been developing a perspective on technical innovation that would be worth exploring for empirical applications, even if I am not the one to do this.

Saturday, February 17, 2018

Marx Versus Classical Economics

Marx can be read as both a continuation and a critique of classical economics. A not-too-radical reading might emphasize his claim to find distinctions in economic theory glossed over by classical economists such as Adam Smith and David Ricardo. According to Marx, classical economists (as opposed to vulgar economists such as Frédéric Bastiat, Jean-Baptiste Say, and and Nassau William Senior) penetrated beneath surface phenomena to reveal the anatomy of capitalism. A more radical reading questions the soundness of the classical theory, while historicizing its emergence as a necessary illusion. The spokesmen for the emerging and progressive capitalist class sought for a theory justifying their opposition to aristocrats and the ancien régime. And classical economics was that theory.

This post presents three distinctions offered in the first, less radical reading. Marx had great respect for classical economists. I do not think he was always fair to them, insofar as he accused them of error by reading muddle into them for not seeing his new ideas. In this post, I do not document this charge by citing specific passages in, for example, Theories of Surplus Value. So more work would need to be done to extend this from a mere blog post. (This 8 January 1868 letter from Marx to Friedrich Engels is apposite here.) I also put aside the transformation problem here.

First distinction: between labor and labor power. Marx distinguishes between the capability of a member of the proletariat to work under the direction or control of a capitalist and the work done under that direction. The former is a commodity, labor power. The latter is the use value of that commodity, that is, labor. Both Marx and Ricardo treated labor power, like all commodities, as representing a certain quantity of embodied labor, namely, the labor value of the commodities necessarily consumed by the laborers, taking as given certain conventions about the hours and severity of work, the standard of living of the workers, the size of their families, which members were expected to work, and so on.

Without this distinction, Ricardo writes about such nonsense as the labor value of labor. (I need a direct quote here.) Marx argues that Ricardo is also unable to explain why capitalists are able to regularly generate profits. I suppose one could expand on this to analyze some of the evident difficulties in understanding Ricardo.

Second distinction: Between surplus value and profits, rent, and interest. Surplus value, for Marx, is the value added by labor not paid out in wages. It is an abstraction, akin to (some of) Ricardo's profits before his chapter on rent. Marx focuses on surplus value in the first volume of Capital. Surplus value is manifested at a more concrete level in the form of profits, rent, and interest on financial instruments. Would Ricardo's work be better if he had a separate label for surplus value?

Third distinction: Between prices of production and labor values. William Petty, Adam Smith, and David Ricardo all have a theoretical conception of market prices and natural prices. Natural prices are centers of gravity, in some sense, around which market prices fluctuate. Marx offered a trichotomy of market prices, prices of production, and labor values. The price of production, sometimes called the cost price, is Marx's equivalent for Smith and Ricardo's natural value. Marx can criticize passages in the classical economists for confusing prices of production and labor values. (A further confusion is that between the labor commanded by and the labor embodied in a commodity.)

I conclude with noting some complications not to be found in the above schematic divisions. In speaking of the labor value of labor power, I am implicitly assuming that all wages are saved, and that wages are paid in commodities. But some workers, especially those deemed skilled, are able to save, even over and above what they need for a conventional retirement. And wages are paid in money, with the general level of prices of wage goods only determined after a bargain with workers has been struck.

In talking about surplus value, I have ignored the possibility of profits on alienation. This case has to be considered in a complete taxonomy of capital. Traders and speculators look for the possibility of bargains, of buying low and selling high. Both classical economists and Marx were aware of this possibility.

In speaking of labor values and prices of production, I seem to be assuming that all firms in an industry use the same processes and have the same costs. But Marx looks at variations in such processes. (I am never sure whether the processes that Sraffa takes as given should be the best practice or an average process. Perhaps, which is correct might vary among industries.) Finally, one might add a fourth distinction in Marx's theory of absolute rent, which is not to be found in the classical economists.

Thursday, February 15, 2018

Another Way Reswitching Can Appear

Figure 1: Wage Curves for a Reswitching Example
1.0 Introduction

This post illustrates another fluke case. In this example economy, two techniques exist for producing a net output of corn. The wage curves for the two techniques have two switch points. One switch point is on the wage axis, corresponding to a rate of profits of zero. The other is on the axis for the rate of profits, corresponding to a wage of zero.

This example is a fluke in two ways. In the jargon I have been inventing, it is simultaneously a pattern across the wage axis and a pattern over the axis for the rate of profits. It differs from this previous example in that the switch points in both patterns arise for the same pair of techniques. In my jargon, it is a global pattern.

I created this example by simplifying and perturbing this one.

2.0 The Model

As usual, managers of firms know of a number of production processes (Table 1). A single commodity - a ton iron, a ton steel, or a bushel corn in the example - is the output of each process. Each process lasts a year and exhibits constant returns to scale. Inputs are defined in physical units, as indicated in the column for the iron-producing process. All inputs are used up in production; there is no fixed capital or joint production.

Table 1: The Technology for a Three-Industry Model
Labor1/3 Person-Yr.1/20.0616280.420472
Iron1/6 Ton1/20010
Steel1/200 Ton1/400.070079
Corn1/300 Bushel1/30000

Two techniques are available. The Alpha technique consists of the iron-producing process, the steel-producing process, and the corn-producing process labeled Alpha. The Beta technique consists of same iron-producing and steel-producing processes, with the corn-producing process replaced by the one labeled Beta.

The choice of technique in a capitalist economy is assumed here to be based on cost-minimization for prices of production. Prices of production, for each technique, are characterized by a system of three equations in which the same rate of profits is earned in all three industries, for the processes comprising the technique. I assume that labor is advanced, and wages are paid out of the surplus. And I take a bushel corn as the numeraire.

Under these assumptions, one can draw the wage curve for each technique, as in Figure 1. The outer frontier of the wage curves illustrates the cost-minimizing technique. In the example, the Beta technique is cost-minimizing whatever the distribution of income. It is not uniquely cost-minimizing, however, for the switch points. In the two cases of a zero rate of profits and a wage of zero, any linear combination of the two techniques is cost-minimizing.

3.0 Conclusion

Suppose the coefficients of production for the corn-producing process in the Alpha technique were slightly higher. Then no switch points would exist, and the Beta technique would be uniquely cost-minimizing, whatever the distribution of income between wages and profits. The coefficients in the example illustrate a boundary case, just as technical progress creates a situation where prices of production arise for a case of reswitching. If technical progress were to decrease the coefficients of production for the Alpha process, the switch points would be closer together and further from the axes. It might be that what I am now calling a reswitching pattern might never occur. Some other processes for producing iron or steel might supplant the ones in the example, like in this previous example.

Saturday, February 10, 2018

Books That Seem Interesting

I am thinking of maybe buying one of these:

I already have many books on which I am behind, for instance, Anwar Shaikh's Capitalism. I suspect the Penrose biography will strike me like Adelman's biography. I've read some of the economics the subject produced, but did not know about the Nazi-fighting.

Saturday, January 27, 2018

A Four-Technique Pattern

Figure 1: Partition of the Parameter Space
1.0 Introduction

I here provide some notes on a perturbation of an example from Salvadori and Steedman (1988).

Consider an economy in which n commodities are produced in n industries. In each industry, a single commodity is produced from inputs of labor and the services of previously produced capital goods. Suppose the technology can be represented in each industry by a continuously-differentiable production function. The wage-rate of profits frontier for such a model does not contain any switch points. In other words, for each feasible rate of profits, a single technique is cost minimizing. Nevertheless, the cost-minimizing technique varies continuously with the rate of profits. Furthermore, the process associated with the cost-minimizing technique in each industry also varies continuously with the rate of profits.

Suppose, instead, that the processes in each industry were represented by a set of fixed-coefficient processes, instead of a smooth production function. What would hold in a discrete model that is in the spirit of the neoclassical model? I suggest that at each switch point on the frontier, 2n wage curves would intersect. In a model with two produced commodities and two processes available in each industry, four wage curves would intersect at the single switch point. With three produced commodities, eight wage curves would intersect. The natural properties for a neoclassical model - if that is what this is - are flukes to several degrees.

I do not necessarily claim anything revelatory from the details of this post. I am testing the applicability of my pattern analysis by trying it out for various examples. Although you cannot tell from my presentation, the graphs I draw rely less on numerical approximations than in many of my earlier examples. This example is the first I have seen where a pattern with a co-dimension of two or higher happens to form a one-dimensional locus (curved line) in the two-dimensional slice of the parameter space I graph. Salvadori and Steedman could have varied their example in an infinite number of ways and still had an example where all processes varied at a switch point.

2.0 Technology

I make my usual assumptions about technology. At a given point in time, managers of firms know of a number of production processes (Table 1). A single commodity - a ton iron or a bushel corn in the example - is the output of each process. Each process lasts a year and exhibits constant returns to scale. Inputs are defined in physical units. For example, labor inputs are specified in terms of person-years per ton iron output or per bushel corn output. All inputs are used up in production; there is no fixed capital or joint production.

Table 1: The Technology for a Two-Industry Model
Labor1 e1 - σ t2 e1 - φ t12
Corn(2/3) e1 - σ t(1/2) e1 - φ t00

To produce a self-sustaining net output with this technology, both iron and corn must be produced. Four techniques can be defined with this technology (Table 2).

Table 2: Techniques in a Two-Commodity Model
Alphaa, c
Betab, d
Gammaa, d
Deltab, c

I have defined the technology such that coefficients of production decrease with time in both processes for producing iron. The rate at which they decrease differs between the two processes. A more general case would allow for technical process in each of the processes for producing corn.

3.0 A Temporal Path

I first consider the variation with time of prices of production for a special case. Consider:

σ = φ = 1

I make the usual assumptions for prices. Relative spot prices are stationary, such that the same rate of profits is earned in both industries if the technology at a given point of time had prevailed over the year. I assume labor is advanced, and wages are paid out of the surplus at the end of the year. A bushel corn is taken as the numeraire. Supernormal profits cannot be made for either process comprising the chosen technique(s). No process in use incurs extra costs.

Figure 2 shows how cost-minimizing techniques, the maximum rate of profits, and switch points vary with time. In the region label 1, the Beta technique is cost-minimizing for all feasible rates of profits. The Gamma technique is cost-minimizing for high wages and low rates of profits in Reqion 2. A single switch arises, where wage curves for the Beta and Gamma techniques intersect on the frontier. In the language of the technical terminology I have been introducing, the boundary between Regions 1 and 2 is a pattern across the wage axis. Other patterns are labeled in the diagram.

Figure 2: Variation of Switch Points with Time

When t = 1, this model reduces to Salvadori and Steedman's example. A single switch exists, with a rate of profits, r0, of 20 percent and a wage of (1/5) bushel per person-year. The wage curves for all four techniques intersect at the switch point. I call the boundary between Regions 5 and 7 a four technique pattern.

I argue that a four technique pattern is of co-dimension two, in my jargon. Each pattern is defined for a switch point. So, in a pattern, at least two wage curves intersect at a switch point:

wα(r0) = wγ(r0)

The co-dimension is the number of additional conditions that must be satisfied for the pattern. Here are two more conditions:

wβ(r0) = wδ(r0)
wα(r0) = wβ(r0)

In this example, for any switch point between the Alpha and Beta techniques, all processes are cost-minimizing. Thus, all techniques are cost-minimizing at such a switch point. For any set of parameters (σ, φ, t) at which there exists a switch point on the frontier between Alpha and Gamma and between Beta and Delta, all techniques are cost-minimizing. In the example, the first two conditions imply the third because of the processes of which the techniques are composed. I think this implication does not hold in general, for all technologies. So I think the definition of a four technique pattern must include three equalities.

4.0 Partition of the Parameter Space

The above analysis can be generalized, to consider any combination of (σ t) and (φ t). Figure 1, at the top of the post, partitions the parameter space into seven regions. In any given region, the switch points and the wage curves along the frontier do not vary qualitatively. (Maximum wage, maximum rate of profits, and rate of profits for switch points may vary.) Table 3 lists the switch points and wage curves along the wage frontier, for each region.

Table 3: Cost-Minimizing Techniques
RegionSwitch PointsTechniques
2Between Beta & GammaGamma, Beta
4Alpha & GammaAlpha, Gamma
5Alpha & Gamma, Beta & GammaAlpha, Gamma, Beta
6Beta & DeltaDelta, Beta
7Alpha & Delta, Beta& DeltaAlpha, Delta, Beta

As an aid to visualization, I present some specific configuration of wage curves. Consider the point in the parameter space that is simultaneously on the boundary of Regions 1, 2, 5, 6, and 7. At this point, all techniques are cost-minimizing for a rate of profits of zero. It is simultaneously a four-technique pattern and patterns across the wage axis. Figure 3 shows the wage curves in this case. For feasible positive rates of profits, the Beta technique is uniquely cost-minimizing.

Figure 3: Patterns over the Wage Axis

Figure 1 shows loci for four wage patterns intersecting at the point in the parameter space with wage curves illustrated above. Since six pairs of (unordered) techniques can be chosen from four techniques, one might think that six wage patterns should intersect at this point. But I am only defining patterns for switch points on the frontier. To illustrate, consider figure 4, which shows wage curves for a point in Region 5. The wage curves for the Gamma and Delta techniques intersect on the wage axis. Neither, however, are cost-minimizing here; the Alpha technique is cost-minimizing for a rate of profits of zero.

Figure 4: Wage Frontier in Region 5

Region 7 is the other region in three techniques are cost-minizing along the wage frontier. Figure 5 illustrates Region 7. For this particular set of parameters, the wage curves for the Gamma and Delta techniques are tangent at a point within the wage frontier. As far as I can tell, no reswitching patterns arise in this example, for switch points on the frontier.

Figure 5: Wage Frontier in Region 7

It is also the case that if one extends Figure 1 to the right, the locus for the four-technique pattern never ends. There is not some set of parameter values where the wage curves for all techniques intersect at the maximum rate of profits.

In a perturbation of the example, one can find a set of parameters at which the wage curves for all four techniques intersect at a switch point for a rate of profits of zero. And the parameters can be varied such that the rate of profits for a switch point for all four techniques can be any positive rate of profits.

  • Salvadori, Neri and Ian Steedman. 1988. No Reswitching? No Switching! Cambridge Journal of Economics, 12: 481-486.

Wednesday, January 24, 2018

From Odo's Prison Letters

For we each of us deserve everything, every luxury that was ever piled in the tombs of the dead kings, and we each of us deserve nothing, not a mouthful of bread in hunger. Have we not eaten while another starved? Will you punish us for that? Will you reward us for the virtue of starving while others ate? No man earns punishment, no man earns reward. Free your mind of the idea of deserving, the idea of earning, and you will begin to be able to think. -- Ursula K. Le Guin (21 October 1929 - 22 January 2018)