Exemplars of Neoclassical Economics

1.0 Preamble

Some mainstream economists defend themselves from criticisms of neoclassical economics by asserting that the mainstream is not neoclassical. They have all these recent innovations, distinguishing modern economics from archaic neoclassical economics. I judge some of these putting these claims forward to be dishonest, ignorant of the history of economics.

Anyways, I consider the following books, not all of which I have read, to be important developments of neoclassical economics:

• William Stanley Jevons (1871) The Theory of Political Economy
• Carl Menger (1871) Principles of Political Economy
• Leon Walras (1874, 1877) Elements of Pure Economics
• Alfred Marshall (1890) Principles of Economics
• Philip Wicksteed (1910) The Common Sense of Political Economy
• A. C. Pigou (1920) The Economics of Welfare
• Lionel Robinson (1932) An Essay on the Nature and Significance of Economic Science
• Edward Chamberlin (1933) Theory of Monopolistic Competition
• Joan Robinson (1933) The Economics of Imperfect Competition
• J. R. Hicks (1939, 1946) Value and Capital
• Paul A. Samuelson (1947) Foundations of Economic Analysis
• Gerard Debreu (1959) Theory of Value

I rely on English translations, with a bias towards books originally written in English. I also am going to be Whiggish in trying to briefly summarize some elements in these books. The level of mathematical abstraction and formalism increased during this time.

2.0 Selected Neoclassical Doctrines

Walras presents a sequence of models of static equilibrium. He worries about the existence and uniqueness of equilibria, as well as their optimality properties. One can read him as not attempting to abstractly describe existing capitalist economies, but an impossible utopia. He tries to understand how an economy can approach equilibrium, but his solution prohibits false trading and cannot be set in historical time.

Menger does not set out his theory with the differential calculus. It can be read as much more like an attempt to set out a Linear Programming approach in words. I have never absorbed his approach to utility theory, but it seems to postulate a structure to preferences more than that given by a continuous total order. I do not think the latter is considered essential to the Austrian school. The former was not seen so at the time. Wicksteed's emphasis on opportunity cost is seen as Austrian, even though he sets out, in words, epsilon-delta arguments for applications of the calculus to economics.

Marshall is not what I expected. He has a big picture view of economic development. Maybe his theory, when formalized, does not support an analysis of growth and development. I can see supply and demand arguments in various runs and other elements of the microeconomic textbook partial equilibrium models. But I see the textbook treatment much clearer in Chamberlin and Robinson's books. I gather they had to argue that marginal revenue curve was not an innovation of theirs. But the theory of market forms was codified by them.

Robbins is widely cited for the definition of (neoclassical) economics as the study of choices that allocate scare resources among alternative ends. These are not his exact words, and you can find similar definitions going back to, for example, Jevons. I guess it is an irony of history that Robbins set out his definition during the Great Depression, when neither labor nor capital equipment was scarce.

The books I list by Hicks and Samuelson are widely considered foundational for post-war mainstream economics. Hicks was introducing many elements of continental economics, particularly from Pareto, into Anglo-American circles. (Maybe my list above should include Knut Wicksell.) For example, consider the insistence that utility must be ordinal. He also justified Walras' law with an approach based on counting equations. Hicks also presents a model of temporary equilibrium. Here we see another kind of dynamics, which cannot be set in historical time. Spot markets clear, based on agents expectations and plans. Hicks talks about the elasticity of expectations, but does not formally model either adaptive or rational expectations. As I understand it, Samuelson has an approach to dynamics more like the Walrasian tatonnement, which still cannot be set in historical time.

I end my list with Debreu, just to emphasize that topological arguments imported from Bourbaki can still be neoclassical economics. Also, part of my point is that much of mainstream economics is not of recent origin. So I do not want to go on.

3.0 Mainstream Non-Neoclassical Doctrines

I guess I might mention some approaches in mainstream economics that one could argue are not neoclassical. Game theory; behavioral economics, including prospect theory; models of asymmetric information; and the supposed recent empirical revolution spring to mind.

I seem to recall critiques of game theory and behavioral economics as sharing the neoclassical defect of focusing on monads, not considering fully socialized individuals. This may have something to do with how their introduction into economics was accommodated to existing doctrine. In game theory, one can make such an argument, I guess, by contrasting von Neumann and Morgenstern's emphasis on social norms in their model of cooperative equilibria with (refinements of) Nash equilibria. Is not this a major point of Mirowski's Machine Dreams?

As I recall, Stiglitz, in his Nobel acceptance speech, explicitly contrasts asymmetric information with neoclassical economics. But do not information problems go along, to some extent, with the analysis of externalities and the explanation, by the Chicago-school economist Ronald Coase, of the existence of firms in problems of transaction costs? Somewhere (in G. M. Hodgson's work), I have seen it asserted that neoclassical economics is consistent with information problems, as long as they are not too severe.

Work by economists on, say, natural experiments or big data just seems mostly orthogonal to the questions touched on in this post. An examination of Google searches for evidence of racism in the 2008 United States presidential election does not seem to be about the validity of neoclassical economics.

Anyways, the compatibility of the supposed pluralism of existing mainstream economists with their shunning of traditional heterodox schools is too much to go into at the end of a post.

4.0 Conclusion

Clearly, neoclassical economics includes "dynamics", less than perfect competition, and externalities, for example. I can take seriously some of those who struggle to put more up-front emphasis on these elements in introductory teaching (for example, David Colander). I can see why some might want to take a pragmatic approach to argue for less market fundamentalist policy, taking mainstream economics as given. I do not take issue with those, like Franklin Fisher, who have tried to address fundamental flaws in neoclassical economics like its inability to allow for false (out-of-equilibrium) trading. On the other hand, I can see the desirability of starting from heterodox approaches in trying to explore actually existing capitalist economies and their problems.

Comments on Yoshihara and Kwak on Sraffian Indeterminancy

1.0 Introduction

Yoshihara and Kwak (henceforth YK) presented a paper, on Sraffian indeterminacy, at the last annual meeting for the American Economic Society. I want to register my qualified disagreement.

2.0 Yoshihara and Kwak against Mandler

YK are arguing against Michael Mandler. In a 1999 paper, a book, and a series of papers since, Mandler has been criticizing Sraffa and his followers. In Mandler's reading, Sraffa argues that, in neoclassical theory, the distribution of income is generically indeterminate. That is, for any long-period equilibrium solution, one can find another solution as nearby as you want. Or, in still other words, equilibrium solutions are a continuum in some space. In much simpler terms, any distribution and prices along the wage-rate of profits curve is a long-period equilibrium in a circulating capital model with a single technique. Mandler says that Sraffa is more-or-less mistaken.

YK say that Mandler is correct if one confines oneself to stationary equilibria. But, if one considers steady states, with a not-necessarily zero, endogenous rate of growth, then indeterminate equilibria are generic.

3.0 Clarifications and Caveats

I should offer some clarifications and caveats. First, Mandler's claim is consistent with multiple, non-unique equilibria. Equilibria are not indeterminate, as long as the number of equilibrium is finite or, I guess, at most countable. So, if I produce a model in which several points on the wage frontier are neoclassical equilibria, I am not offering a counter-example or disproof of Mandler's claim that Sraffian equilibria are determinate.

Second, Mandler has a caveat. In particular, he argues indeterminancy arises in a short-run model with technology modeled as Leontief matrices. The capital goods that exist at the start of any period are the result of production in the previous period. If they were taken as given, some might be in excess supply with an equilibrium price of zero. And those not in excess supply would have a determinate price. But there is a boundary, just before capital goods are in excess supply price. There, equilibrium prices would range from zero to some upper bound. For strategic reasons, managers of firms have an incentive to produce just this quantity of capital goods.

Third, YK are arguing in the framework of a model of overlapping generations. The production technology is specified by a Leontief matrix. The model is extended to include utility maximization by households. Each household must decide how much and what to consume out of wages and what to consume, at the end of a second period, out of retirement savings. I think assumptions that households only live for two periods and that they must work the first period and be retired the second period are inessential. Their results are most likely consistent with labor supply being determined by including a preference for leisure in the utility function. And one can have households lasting more than two periods, with more than two generations being included in the demand for consumer goods at any period.

4.0 My Objections

4.1 Objections to Mandler's Reading of Sraffa

I have not read Sraffa for decades, if ever, in line with Mandler. Sraffa does not mention utility functions. And he does not model quantity equations, although I find it natural to add a system of steady state growth to Sraffa's model. Sraffa says he intends his book to provide a foundation for a critique of (neoclassical?) economics, but he certainly presents problems of interpretation for stating what that critique is.

I think of Sraffa as presenting an open model. I guess one can say his solutions are indeterminate, but I do not see him as saying that a neoclassical closure is necessarily indeterminate.

Rather Sraffa shows that one can still model prices and distribution without any reference to subjective, neoclassical theory. He presents a model that can be closed in various ways. Neoclassical theory is only one approach out of others. Furthermore, Neoclassical economists do not seem to have any theoretical foundation for their vision of prices as indices of relative scarcity, as reflecting the result of an overriding principle of substitution.

4.2 An Objection to Yoshihara and Kwak

As I understand it, YK define a steady-state equilibrium by a stationary vector of relative prices, wage, rate of profits, a vector of gross outputs, and a rate of growth. The gross outputs and the rate of growth specify a time path for gross outputs and employment, all components of which grow at the steady state rate.

One can vary the rate of growth continuously in a certain range. Since the parameters of the household utility functions are given, the steady state distribution of income and, consequently, prices must vary too. Voila, steady state equilibrium are indeterminate.

I guess this is consistent with an extension of Mandler's concept of indeterminancy. But it does not seem in the spirit of neoclassical economics, which is the about the allocation of given resources. In my excursions into models of overlapping generations, I have always taken the rate of growth of the population as given, that is, exogenous. I have seen, at least, that if one varies certain parameters in the utility function, the stationary state equilibrium varies continuously. By labeling this a model of endogenous time preferences, have I proven Mandler wrong, even for stationary state equilibria? Do not these disproofs, if that is what they are, even work for aggregate Cobb-Douglas production functions?

5.0 An Empirical Issue?

I am not at sure these are at all questions that can be settled by mathematical modeling. Sraffa presents an open model. Why feel obligated to close it with a formal model, much less with neoclassical assumptions of utility maximizing?

Instead, one can say that Sraffa has presented a model where one can find a place for political forces to impact the distribution of income, in all runs. One can use Sraffa as a justification for, for example, looking at the impact of the policies of the Federal Reserve on income distribution, without being required to create a formal model at the level of abstraction of Sraffa's book.

Likewise, isn't the question of whether the size of the workforce varies endogenously also an empirical question? Offhand, I think of how the labor force participation rate has varied over the last decade, with the advent of the Global Financial Crisis; how estimates of the Non-Accelerating Inflation Rate of Unemployment (NAIRU) have fallen with unemployment over decades; and the increased participation of women in the workforce during World War II as cases to pursue.

REFERENCES

• James K. Galbraith (2000). Created Unequal: The Crisis in American Pay, University of Chicago Pay
• Michael Mandler (1999). Sraffian Indeterminancy in General Equilibrium. Review of Economic Studies 66: 693-711.
• Frank Hahn (1982). The Neo-Ricardians. Cambridge Journal of Economics 6: 353-374.
• Stephen A. Marglin (1984). Growth, Distribution, and Prices. Boston: Harvard University Press.
• Naoki Yoshihara and Se Ho Kwak (2017). Sraffian Indeterminacy in General Equilibrium Revisited. Proceedings of the American Economic Society

Structural Economic Dynamics with a Choice of Technique in General

Many - not all - of my recent numerical examples have a certain abstract pattern:

1. At the start of the time under consideration, one technique is uniquely cost minimizing, for all feasible rates of profits.
2. Coefficients of production decline or some markups over the normal rate of profits vary.
3. A fluke switch point appears.
4. Switch points move along the wage frontier, and interesting phenomena occur. These can be other fluke switch points. Reswitching, the recurrence of techniques, capital-reversing, the reverse substitution of labor, or process recurrence might arise for some time.
5. Eventually, these interesting phenomena disappear, and another technique is uniquely cost minimizing, for all feasible rates of profits.

I have not been looking at random technology. The occurrence of a fluke switch point is not surprising in my examples. Neither is some of the phenomena mentioned in (4). I have been deliberately creating examples to highlight some of these possibilities. But I have often found a second or more fluke switch points arising that I did not expect. I have also created examples in which one technique gets replaced with another, but in which reswitching, etc. do not occur.

My program remains unfinished. In these posts and papers, I have suggested the possibilities of some proofs and additional fluke switch points. I have yet to even begin considering how the changes in the wage frontier I have been investigating might manifest themselves in the movement of market prices. I could do more with markups varying among industry. I have yet to provide examples with land and fixed capital, where the wage frontier is not the appropriate tool for analyzing the choice of technique.

A Reswitching Pattern With A Continuum Of Techniques

Recurrence Of Techniques Without Switch Points

I have built on my previous post in a writeup:

Abstract: In certain models of commodities produced by means of commodities, the choice of technique is analyzed by the construction of the wage frontier. This article presents a numeric example of a continuum of wage curves tangent at a switch point. Technological progress leads to the recurrence of techniques. No switch points then exist, but the cost-minimizing technique varies continuously along the wage frontier. Further progress leads to the disappearance of the recurrence of techniques and, eventually, a single technique becoming cost-minimizing for all feasible rates of profits.

Economists Critical Of Mainstream Economics Not Going Into Academia

Here are three books I have read:

• Paul Ormerod (1994). The Death of Economics. St. Marin's Press.
• Stanley Wong (1978, 2006). Foundations of Paul Samuelson's Revealed Preference Theory: A Study by the Method of Rational Reconstruction. Routledge
• J. E. Woods (1990). The Production of Commodities: An Introduction to Sraffa. Humanities Press.

These authors have two things in common. All three were critical of some aspects of mainstream economics. And they all ended up in business. Looking at the blurbs on their books, I see some spent more time in academia than I recalled.

I wonder if one can find something like a trend. Are there many economists that have come out of well-regarded economics department and had too critical a mind? And they ended up either in business or in departments less well-regarded? Maybe Thomas Palley (Yale?) fits in here.

Of course, there is another phenomenon of engineers, mathematicians, and scientists looking at economics from the outside. Mirowski is good on this theme. John Blatt is somebody Post Keynesians might cite here. Notice, this goes back well before the recent enthusiasm for econophysics.

A Generalized Reswitching Pattern

Figure 1: Switch Points Varying with Time

1.0 Introduction

This post presents a perturbation of a fluke switch point. At this switch point, the wage curves for four techniques are tangent. In the jargon I have been inventing, this is another four-technique, local pattern. In other words, a perturbation of appropriately selected parameters - for example, coefficients of production - changes the sequence of wage curves and switch points on the wage frontier. The perturbation can be viewed as the result of technical progress. When I worked the example, I was surprised to find some other fluke cases.

The numeric example is an instance of the Samuelson-Garegnani model. Of interest to me, is a generalization to a continuum of techniques with wage curves tangent at the switch point. A perturbation leads to an example with no switch points, but the cost-minimizing technique varies continuously along the wage frontier, and techniques recur. So this generalization will have the structure of an example in Garegnani (1970). Kurz and Salvador (2003) later simplified this famous example. In some sense, I am offering a further simplification. But, perhaps, my example is more complicated along other dimensions, insofar as my pattern analysis is original.

2.0 Technology

In this economy, a single consumption good - called corn - is produced from inputs of labor and a specified grade of iron. The grade of iron is indexed by v, u₁, u₂, and u₃. Each grade of iron is itself produced from inputs of labor and that grade of iron. Table 1 shows the processes available, at each point in time, for producing iron. Similarly, Table 2 defines the processes available for producing corn. This is a circulating capital model. The iron inputs are totally used up in a single production period.

Table 1: The Technology in the Heterogeneous Iron Industry

Input	Process
	v	u1	u2	u3
Labor	(2/5)	(5/18)e-(t - 1)σ1	(49/360)e-(t - 1)σ2	(2/45)e-(t - 1)σ3
v Iron	(1/5)	0	0	0
u1 Iron	0	(1/4)e-(t - 1)σ1	0	0
u2 Iron	0	0	(13/40)e-(t - 1)σ2	0
u3 Iron	0	0	0	(2/5)e-(t - 1)σ3

Table 2: The Technology in the Corn Industry

Input	Process
	v	u1	u2	u3
Labor	2	(20/9)	(23/9)	(26/9)
v Iron	1	0	0	0
u1 Iron	0	1	0	0
u2 Iron	0	0	1	0
u3 Iron	0	0	0	1

Four techniques for producing a net output of corn exist. Each technique consists of a process for producing iron of a specific grade and a process for producing corn with that grade of iron. For my notes when extending this example to a continuum of techniques, I want to note the following restriction and relationships among coefficients of production:

(1/5) ≤ a_1,1(u, 1) < (1/2)

a_0,1(u, 1) = (10/9)[1 - 2 a_1,1(u, 1)]²

a_0,2(u, 1) = (10/9)[1 + 4 a_1,1(u, 1)]

3.0 Prices

Suppose the technique defined by the u grade of iron is in use. Consider the associated prices of production, for the period of production ending at time t. Let r be the rate of profits, w_u(r, t) be the wage, and p_u(r, t) the price of u-grade iron. Prices are production satisfy the following system of two equations:

p_u(r, t) a_1,1(u, t) R + a_0,1(u, t) w_u(r, t) = p_u(r, t)

p_u(r, t) R + a_0,2(u, t) w_u(r, t) = 1

where:

R = 1 + r

A bushel corn is the numeraire.

One can solve this system for the wage and the price of corn, each as a function of the rate of profits and time. The wage, as a function of the rate of profits, is called the wage curve for the technique. A different wage curve is defined for the technique defined by each grade of iron. The wage curve for the v-grade of iron does not shift with time.

4.0 Choice of Technique

The wage curves, for each of the techniques defined by a grade of iron, can be plotted on the same graph. This graph depicts wage curves and the wave frontier at a given point in time. The wage frontier is the outer envelope of the wage curves. The technique(s) that contribute their wage curve(s) to the frontier are cost minimizing for the corresponding rate of profits or wage.

4.1 The Wage Frontier at t = 1

Figure 2 shows the wage frontier at t = 1. The technique defined by v-grade iron is cost-minimizing for all feasible rates of profits. All four techniques are cost-minimizing at the single switch point. All four wage curves are tangent at the switch point. This is a fluke.

Figure 2: Wage Curves Tangent at Switch Points

In Figure 2, I have indicated the rate of technical progress for the three techniques defined by u₁, u₂, and u₃. But, with the way I have defined technical progress, these rates do not matter for prices of production at time t = 1. Furthermore, for any time less than unity, the wage frontier is the same. The wage curve for v-grade iron does not shift, and the technique defined by v-grade iron is uniquely cost-minimizing for all feasible rates of profits. The story is different for as time goes on after t = 1.

4.2 The Shift in Wage Frontier when Technical Change is Faster for Smaller a_1,1(u, 1)

First, consider the case when σ is smaller for a larger index u for the grade of iron. Figure 3 graphs such a case, for a time larger than t = 1. This is an example of reswitching, between the techniques defined by v-grade and u₁-grade iron. The wage curves for the techniques defined for u₂-grade and u₃-grade iron appear on the frontier only at t = 1 and only at the switch point. Otherwise, this is a perturbation analysis, for these rates of technical progress, that yields a traditional reswitching example.

Figure 3: A Reswitching Example

4.2 The Shift in Wage Frontier when Technical Change is Faster for Larger a_1,1(u, 1)

I created this example more with this case in mind. In obvious notation, define the rates of technical progress by:

σ_u = (1/10) a_1,1(u, 1)

Figure 4 graphs the wage frontier shortly after t = 1. The wage curves for the techniques defined by v-grade, u₁, and u₂ each appear in two separate regions on the wage frontier. The single switch point has become six switch points. Perturbation of the coefficients of production for the fluke switch point has yielded an example of the recurrence of techniques.

Figure 4: Recurrence of Techniques

Around the three switch points at the larger rates of profits, a higher wage is associated with the adoption of a cost-minimizing technique where more labor is employed per unit of the consumption good produced. When will mainstream economists stop telling lies to students about price theory and the logic of minimum wages?

Figure 5 graphs the wage frontier at the following time:

t = 1 - 40 ln(4/5)

This is an example that is simultaneously a pattern across the wage axis and over the axis for the rate of profits. The technique defined by v-grade iron, and the associated switch points, is disappearing from the wage frontier. I did some work to previously create such a global pattern. I do not know what specific, presumably special case conditions, I have imposed to make this pattern come about. These numeric examples keep

Figure 5: Switch Points on Both Axes

Figure 6 graphs the wage frontier at an even later point in time. Three switch points appear on the frontier. Three wage curves intersect at the switch point at the larger rate of profits. This is what I call a three-technique pattern. The wage curve for the u₂-grade iron is disappearing from the wage frontier.

Figure 6: A Three-Technique Pattern

The diagram at the top of this post summarizes my analysis for this case. The rates of profits for switch points are plotted against time. The maximum rate of profits is also shown.

5.0 Conclusion

I have exhibited a numerical example in which four-wage curves are tangent at a single switch point. Technical progress alters certain parameters - that is, coefficients of production - for three of the four techniques. For any time less than the time at which the fluke switch point occurs, no switch point exists. Given a certain simple specification of the rates of technical progress, the switch point breaks up into six switch points for a small increase in time. Three of the four technique recur on the wage frontier. In my jargon, the fluke case is a four-technique pattern. It generalizes the reswitching pattern I have previously defined. I claim that I can create a n-technique generalized reswitching pattern, for any finite n greater than unity. I can also create generalized reswitching pattern with a continuum of wage curves tangent at the single switch point.

My next steps, if I go on, might be to explicitly write up the generalization to a continuum of techniques. I should also find a closed-form for the time at which the above three-technique pattern occurs. (I found it through a bisection algorithm.)

References

• P. Garegnani (1970). Heterogeneous Capital, the Production Function and the Theory of Distribution. Review of Economic Studies, V. 37, no. 3: pp. 407-436.
• Heinz D. Kurz and Neri Salvadori (2003). Reswitching - Simplifying a Famous Example. In Kurz and Salvadori (eds.) Classical Economics and Modern Theory: Studies in Long-Period Analysis Routledge.

Update To A Start On A Catalog Of Switch Point Patterns Of High Co-Dimension

I have been looking at patterns of switch points. A pattern is a configuration of switch points helpful for perturbation analysis for the choice of technique. I am curious how the switch points and the wage curves along the wage frontier can alter with parameters, in a model of the production of commodities. Such a parameter can be a coefficient of production; time, where a number of parameters are functions of time; or the markup in an industry or a number of industries. A normal form exists for each pattern. The normal form describes how the techniques and switch points along the frontier vary with a selected parameter value. Each pattern is defined by the equality of wage curves at a switch point and one or more additional conditions. The co-dimension of a pattern is the number of additional conditions.

I claim that local patterns of co-dimension one, with a switch point at a non-negative, feasible rate of profits can be described by four normal forms. I have defined these patterns as a pattern over the axis for the rate of profits, a pattern across the wage axis, a three-technique pattern, and a reswitching pattern. This post is an update, and continues to examine global patterns, local patterns with a co-dimension higher than unity, and sequences of local patterns. Some examples are:

• 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). This illustrates a pattern of co-dimension two.
• A reswitching example with one switch point being a pattern across the wage axis (another case of a real Wicksell effect of zero). This is a global pattern.
• An example with a pattern across the wage axis and a pattern over the axis for the rate of profits. This is a global pattern.
• A pattern like the above, but with both switch points being defined by intersections of wage curves for the same two techniques. This is a global pattern.
• 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. This gestures towards a global pattern.
• 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). This is a global pattern.
• An example where every point on the frontier is a switch point. This is a global pattern of an uncountably infinite co-dimension.
• Speculation on three sequences of patterns of co-dimension one that result in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
• A switch point for a four-technique pattern (due to Salvadori and Steedman). This is a local pattern of co-dimension two.
• Further analysis of the above example.
• An example of a four-technique pattern in a model with three produced commodities. This local pattern of co-dimension two results in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
• Further analysis of the above example. Two normal forms are identified for four-technique patterns.

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 have used these patterns to tell stories about how technical change or a change in markups (that is, structural economic dynamics) can result in reswitching, capital reversing, or the reverse substitution of labor appearing on or disappearing from the wage frontier.

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.

Workers Benefiting From Increased Markups In Selected Industries

Figure 1: Variation in Switch Points with the Markup in the Iron Industry

1.0 Introduction

I finally use the tools of pattern analysis that I have been inventing to tell a practical story. I build on the example which I began in my previous post.

Workers would be better off if an increase in wages led to greater employment, not less. A long-period change in relative markups among industries can result in firms in some industries obtaining a greater rate of profits at the expense of firms in other industries. But such a change can also create a switch point that exhibits capital-reversing. Around such a switch point, a higher wage is associated with firms adopting a technique of production in which more labor is hired, in the economy as a whole, to produce a given net output. Thus, the change in relative markups leaves workers in a better position to press for a greater share of the surplus product.

2.0 Postulates

In telling this story, I am assuming that possibilities in a simple model - but not too simple - can enlighten us on possibilities in the actual economy. My example is one of an economy in which three commodities (iron, steel, and corn) are produced by means of commodities. I take corn as numeraire and assume wages are paid out of the surplus at the end of the year. Firms have a choice of two processes in each industry for producing the output of that industry. Details are in the last post

Any actually-existing economy cannot ever be expected to be in equilibrium. Nevertheless, I assume that prices of production cast light on tendencies over time for market prices. I realize that this is a contentious claim, especially for Post Keynesians that take issues of fundamental uncertainty seriously.

I assume, for this story, that some sort of long period wage is taken as given when calculating prices of production. It reflects norms about consumption, institutions like how widespread labor unions are, minimum wages, conventions on how bargaining for wages, worker militancy, the policy of the monetary authority, and so on. Some of these influences can be changed, with consequent effects on long period wages and prices of production.

Prices of production also reflect conventions on relative rates of profits. In the example, the rate of profits in the iron industry is 100 s₁ r, 100 s₂ r in the steel industry, and 100 s₃ r in the corn industry. I take it that these conventions can be changed, also with resulting effects on the rate of profits.

3.0 Results and Discussion

Figure 1, at the top of this post, graphs the scale factor for the rate of profits, r, against the markup, s₁, in the iron industry. In drawing this graph, the markups in the steel and corn industries, s₂ and s₃, are taken as unity. The maximum rate scale factor for the rate of profits is also graphed, with the region above it in the graphed labeled as an infeasible region. The thin vertical lines show markups in the iron industry at which four-technique patterns occur.

Switch points between the Delta and Gamma technique appear on the wage frontier in two parameter ranges for the markup in the iron industry. For s₁ between approximately 0.66653 and 1.6195, the switch point between these techniques exhibits capital-reversing. If the markup in the iron industry is slightly below this range, a persisting increasing alters prices of production so that workers pressing wage claims can be more advantageous to them. If the markup in the iron industry is slightly above this range, an increase in the markups in the steel and corn industries can benefit the workers in the same way.

3.1 Selected Examples of Wage Frontiers

I presented three examples of wage frontiers in the previous post. I might as well present two more examples of the wage frontiers here. Figure 2 shows the wage frontier for the value of markups at the point where the switch point between the Gamma and Delta techniques exhibits capital-reversing. The Delta and Theta techniques are only cost-minimizing at the switch point. (One cannot visually distinguish between the wage curves for these two techniques around the switch point.)

Figure 2: The Wage Frontier at a Four-Technique Pattern, with Capital Reversing Appearing

Figure 3 shows the wage frontiers for the value of markups at the point in parameter space where the "perverse" switch point between the Gamma and Delta techniques disappears from the wage frontier. In this illustration, the Alpha and Gamma techniques are cost-minimizing only at the the switch point. For a slightly larger markup in the iron industry, the wage curves for neither the Alpha nor the Gamma technique appear on the wage frontier.

Figure 3: The Wage Frontier at a Four-Technique Pattern, with Capital Reversing Disappearing

3.2 Four Technique Patterns

The graph in Figure 1 demonstrates that at least two patterns over the axis for r and four four-technique patterns arise in this example.

The outer two four-technique patterns resemble one another in some ways. In both cases, processes in two industries are changed, for the (middle) technique that is replaced between the left and the right of the pattern. The Gamma and Epsilon techniques differ in the processes used to produce iron and steel. The same process is used, in both techniques, to produced corn. Similarly, the Gamma and Theta techniques differ in the processes used to produce iron and corn. They share the same process in the production of steel.

The inner four-technique patterns differ from the outer two, but resemble one another. They both show a single switch-point on one side of the pattern being transformed to three switch points on the other side. The wage curves for two new techniques, at least in the region around the switch point, appear on the wage frontier for the parameter values of the markup with the three switch points. I have not previously presented such a pattern. (The structure of the example, in which two processes, but no more than two, are defined for each industry ensure that a three-technique pattern cannot arise in the example. If the wage curves for three techniques intersect in a single switch point, the wage curve for a fourth technique must also go through that switch point as well.)

This example points out the need for normal forms for patterns. I want to formalize the idea that some patterns are topological equivalent, yet differ for others. The presentation of two normal forms for four-technique patterns, which I have only gestured at here, does that.

4.0 By Way of Conclusion

I like how this story combines ideas I take from Kalecki and Sraffa. I do not worry about the labor theory of value, but just take as given that capitalist firms are able to acquire some part, over above what is paid out in wages, of the surplus product.

Appendix

I document that two mainstream economists stated that reswitching has implications for labor markets and income distribution:

"One final and somewhat fanciful remark may be made with reference to this [reswitching] example. Two mixed types of stationary state ... are possible... Both use the same equipment, but the question of ... what income-distribution between labour and capital is fixed, is left in this model for political forces to decide. It is interesting to speculate whether more complex situations retaining this feature are ever found in the real world." -- D. G. Champernowne (1953- 1954)

"By contrast, one who believes technology to be more like my 1966 reswitching example than like its orthodox contrast, will have a more sanguine view about how successful militant power by organized labor can be in causing egalitarian shifts in the distribution of income away from property even in the long run." -- Paul A. Samuelson (1975)

References

• D. G. Champernowne (1953-1954). The Production Function and the Theory of Capital: A Comment. The Review of Economic Studies, V. 21, No. 2: pp. 112-135.
• Paul A. Samuelson (1975). Steady-State and Transient Relations: A Reply on Reswitching. Quarterly Journal of Economics, V. 89, No. 1: pp. 40-47.
• Graham White (2001). The Poverty of Conventional Economic Wisdom and the Search for Alternative Economic and Social Policies. The Drawing Board: An Australian Review of Public Affairs, V. 2, No. 2: pp. 67-87.