Two Systems Thinking Models: Mind Your Ps and Qs

Figure 1: A Market Mediated By Quantity

1.0 Introduction

I have been examining John D. Sterman's textbook, Business Dynamics. Sterman is a chaired professor at the Sloan School of Management and director of the System Dynamics Group at the Massachusetts Institute of Technology (MIT). The System Dynamics Group was founded by Jay Forrester, and the group is continuing research in his tradition.

This systems thinking approach provides tools for visualizing the hypothetical causal relationships and structures of dynamical systems. They show models in which hypothetical causal relationships, the distinction between stocks and flows, and temporal lags can be postulated and displayed. Software for specifying model structures provides capabilities for simulating dynamical behavior. These tools are directed towards managers who may not fully understand complex dynamical systems. The diagrams are intended to package and facilitate informal discussions about models, including desired system states. Simulations for the resulting models give some understanding of possible dynamics.

Sterman's diagrams and associated tools are one approach. Researchers in related disciplines have proposed other visual languages, with varying degrees of formalism for the syntax and semantics of the elements of such diagrams. I think of system block diagrams and the Unified Modeling Language (UML), for instance. Likewise, a number of tools exist (for example, Steve Keen's Minsky system, MathWorks' Simulink, Berkeley's Ptolemy system, and tools supporting Model-Driven Architecture and Model-Driven Development) for processing corresponding system specifications for various purposes.

2.0 "Tell Me What the Wires Do"

I might as well explain a bit about selected components of what Sterman calls Causal Loop Diagram (CLD). CLDs contain curved arrows connecting variable names. The arrowheads in CLDs are annotated with either a plus or a minus sign. Arrowheads indicate causal relations. Suppose an arrowhead points from the variable X to the variable Y:

• Positive Link: If the arrowhead is labeled with a plus sign, Y increases when X increases, all else equal. In other words, ∂Y/∂X > 0.
• Negative Link: If the arrowhead is labeled with a minus sign, Y decreases when X increases, all else equal. In other words, ∂Y/∂X < 0.

A CLD may contain circles with arrows, where each circle contains either the letter B or R, indicating, respectively, either a negative (balancing) or positive (re-enforcing) loop. The dynamical behavior of a system containing a single balancing loop is to approach an equilibrium point. On the other hand, a system containing a single re-enforcing loop exhibits exponential growth. The dynamical behavior of a system containing a combination of interacting balancing and re-enforcing loops, especially if it is non-linear, is more difficult to predict without simulation.

3.0 Two of Three Models

Since Sterman's textbook is directed towards business managers, he provides some examples from economics. In Section 5.5, he presents three models of a single market:

• Demand and supply responding to price (Figure 5-26 in Sterman (2000), Figure 2 below)
• Orders and production respond to queues (Half of Figure 5-27 in Sterman(2000), Figure 1 above)
• Customer base and service quality interact (Other half of Figure 5-27 in Sterman (2000), not shown here)

Figure 2: A Market Mediated By Price

I think Sterman's model of demand and supply mediated by price mixes classical and neoclassical ideas. One should read "demand" and "supply" in Figure 2 as, by an abuse of language, actually referring to the quantity demanded and the quantity supplied. We see that this model postulates that firms increase the quantity supplied for industries in which profits are high, that is, when the price increases more above the cost of production. This is a classical idea, to be found in Adam Smith. The model also postulates that an increase in the quantity demanded puts upward pressure on price. I think how demand is conceptualized in this model, including the role of substitution in consumption, is close to how demand functions are presented in neoclassical textbooks.

Figure 1 shows a model in which firms respond more to increased demand by changes in the level of production, not by changes in price. If price were to be inserted into this model, price would be appropriately modeled by theories of administered, full-cost, or mark-up pricing.

I am not sure I agree with all of Sterman's economic examples. But the above picture of markets fits a Post Keynesian view, articulated by Michal Kalecki, that different microeconomic theories are needed to describe the prices and quantities for markets for raw materials, industrially-produced goods, and services. Do business schools provide a somewhat greater opening for non-neoclassical economics than supposedly leading economics departments?

References

• John D. Sterman (2000). Business Dynamics: Systems Thinking and Modeling for a Complex World, Irwin McGraw-Hill

On "Substitutability"

"[The] validity [of the Cambridge Criticism of neoclassical theory] is unquestionable, but its importance is an empirical or an econometric matter that depends upon the amount of substitutability there is in the system. Until the econometricians have the answer for us, placing reliance upon neoclassical economic theory is a matter of faith. I personally have the faith; but at present the best I can do to convince others is to invoke the weight of Samuelson's authority." -- C. E. Ferguson (1969) [as quoted in Carter (2011)].

1.0 Introduction

In this post, I describe two different meanings of "substitutability", as used in the literature and economists' remarks on the Cambridge Capital Controversy¹.

2.0 Joan Robinson's Criticism

Imagine two island capitalist economies, Alpha and Beta, each in a steady state and with access to the same technology. Suppose for some reason, the distribution of income happens to be different in the two islands. Then the capitalists on the islands will, maybe, have adopted different techniques of production and be producing a different mixture of commodities for final output. Consequentially, the structure of capital goods, both in composition and in quantities, will differ between the two islands.

An (illegitimate) thought experiment is to imagine the distribution of income slowly changing from as it is on one island to the distribution on the other. One might mistakenly consider the capital equipment slowly changing through the composition appropriate to imaginary intermediate islands. This claim ignores the reality of what Joan Robinson called historical time. One is treating a process occurring in time as if it occurring in space, ignoring that past bygones are gone, and assuming no difficulties exist in getting into equilibrium.

Neoclassical² economists frequently ignore the structure of capital equipment and the plans of the entrepreneurs. One meaning of "substitutability" is the assumption that capital goods can be instantaneously taken apart and reassembled to be appropriate for whatever equilibrium is being considered. The tranverse from one equilibrium to another is abstracted from. Robinson satirized this meaning of substitutability by designating the capital good in, say, the Solow-Swan growth model with such names as "ectoplasm", "leets", and "mecanno sets". Post Keynesians, including Sraffians, are generally suspicious of this approach. (Any fans of Austrian school economists want to chime in in the comments?)

3.0 Substitutability and Smooth Microeconomic Production Functions

Another meaning relates to the smoothness of production functions. One might say substitutability exists when derivatives (including, second, third, etc. derivatives) exist for all production functions. That is, substitutability exists in these examples, but not in these ones. (But what would you say about this one, where the cost-minimizing technique varies continuously with the interest rate, and output and each capital good are produced with fixed-coefficients?)

As far as I know, capital-reversing, for example, is consistent with substitutability, in this sense of smooth production functions. I, too, will invoke the weight of Samuelson's authority, even though I reject it in the former case. I would like, however, to see an explicit numeric example.

4.0 Conclusion

I believe C. E. Ferguson was referring to my section 2 meaning of "substitutability". That is, when neoclassical economists claim that Sraffians rely on a lack of substitutability for their critique of neoclassical economics, they should not be objecting to a lack of differentiability of microeconomic production functions.

Footnotes

1. Other usages are ignored in this post. For example, J. R. Hicks' "elasticity of substitution", as used in his mistaken Theory of Wages (1932), is not treated here.
2. As far as I am concerned, "neoclassical" is a meaningful and appropriate word in this context.

References

• Scott Carter (July 2011). C. E. Ferguson and the Neoclassical Theory of Capital: A Matter of Faith, Review of Political Economy, V. 23, N. 3: pp. 339-356

Elsewhere, On Neoclassical Economics

• Noah Smith complains about the supposed overuse of the label.
• Alex Marsh comments.
• Matias Vernengo responds.
• Lars Syll responds, including in pictures. Also, see here.
• I wrote much of the wikipedia article, albeit not the introduction. And some stuff in it I now disagree with. I also wrote much of the Wikipedia article on Classical economics, and the subsection of that article is especially relevant to a Sraffian perspective on neoclassical economics.
• Daniel Kuehn shares some thoughts.
• David Ruccio comments.

Update: Added some links.

Haikus

These have been written by authors who have not acknowledged their authorship. (I have written many of them myself.)

John Maynard Keynes
Created Bretton Woods system
The theory works

Michal Kalecki
Macro with markup pricing
Empirical success

Nicholas Kaldor
Nonlinear business cycle
Generates chaos

Lorie Tarshis
Elements of Economics
Met McCarthyism

Richard Goodwin
Tenureless at Harvard
Among best of the best

John Kenneth Galbraith
The world listened
Not economists

Joan Robinson
Predicted stagflation
Bastard golden age

John Hicks
Renounced IS/LM
Post Keynesianism

Nicholas Kaldor
Defeated Milton Friedman
Endogenous money

Paul Davidson
Explains historical time
Nonergodicity

Post Keynesians
Kicked out of Rutgers
Alfred Eichner died

Wynne Godley
Invents Sectoral Balances
Predicts Crises

The genius of Keen
His splendid model
Remains unrecognized

Post Keynesianism
Destroyed Reinhardt and Rogoff
No surprises here

A Continuous Time System Block Diagram For Nicholas Kaldor

A System Block Diagram For A Business Cycle Model

In this model of business cycles, two state variables, Y(t) and K(t), represent national income and the value of the capital stock, respectively. These state variables are each specified by a differential equation. In the above block diagram, I have adopted a notation from Steve Keen. The triangles in the upper-right and lower right equate the integrals of their inputs, over time, to their outputs. In other words, the following differential equations obtain:

dY/dt = α[I(t) - S(t)]

dK/dt = I(t) - δK(t)

You can compare and contrast this continuous-time representation of a dynamical system with its analogous discrete-time version.

This is a multiplier-accelerator model that allows for the economy to normally be out of equilibrium. An economic interpretation¹ of the model is that entrepreneurs have some sort of common opinion about the level of economic activity they expect in this nation's economy. And they have an opinion about the total value of capital stock that they believe is needed to sustain that activity. When these expectations are realized, this dynamical system is in an equilibrium. The model shows that when the economy has more activity than expected, entrepreneurs tend to increase the capital stock more rapidly, and vice versa for when activity falls below the expected level. This tendency is a non-linear relationship. Maybe, the more extreme the difference between the actual level and the expected level is, the less likely entrepreneurs are to expect the actual level to continue.

Neither interest rates nor prices are modeled here. Such modeling might be justified by the claim that the income effects in the model overwhelm the effects of prices. At any rate, this model does not contain an aggregate production function. Capacity can be operated either above or below the rate that was desired when the capital equipment being evaluated was installed. If the value of the capital stock falls below the expected level, entrepreneurs tend to increase investment, and vice versa for when the value of the capital stock rises above the expected level. (I think of the depreciation of the capital stock shown in the model as an accounting heuristic, not a physical decay.)

I am not putting forth grand empirical claims. To me, this model is of mathematical interest. It illustrates how non-linear economic dynamics can be generated endogenously. A source of continuous external shocks is not needed².

Unlike in the discrete-time case, I do not see how the continuous-time model given here can generate chaos. Trajectories in the two-dimensional state space are smooth, with no gaps. They cannot intersect. So, I think, this continuous-time model can generate cycles, but not strange attractors³. Another difference between discrete-time and continuous-time systems revolves around the details of stability analysis⁴.

Anyways, the graphical specification of the Kaldor model, given in this post, is suitable for numerical exploration in Steve Keen's software, as I understand it.

Footnote

1. As I understand it, mainstream macroeconomists currently reject the rough-and-ready microfoundations I provide here. They insist on formal microfoundations, even though their preferred formal treatments are just nonsense.
2. Some more mainstream economists seem to be willing to make this points in Overlapping Generations (OLG) models. I am willing to explore the mathematics there, despite the absurdity of assuming investment is driven by intertemporal utility-maximization of consumption.
3. The logistic equation is an example of a one-dimensial, discrete-time, chaotic dynamical system. Off-hand, I cannot think of a continuous-time chaotic system with less than three dimensions.
4. In discrete-time systems, one analyzes the stability of a fixed point by analyzing whether the eigenvalues of the system, linearized around the fixed point, are inside or outside the unit circle in the complex plane. In a continuous-time system, one looks to see if the eigenvalues are to the left or the right of the complex axis, if I recall correctly.

Marx On Ricardo

Karl Marx wrote a lot about David Ricardo's economics. Here is some of what he had to say in Theories of Surplus Value:

Ricardo starts out from the determination of the relative va1ues (or exchangeable values) of commodities by “the quantity of labour”.  (We can examine later the various senses in which Ricardo uses the term value.  This is the basis of Bailey’s criticism and, at the same time, of Ricardo’s shortcomings.)   The character of this “labour” is not further examined, If two commodities are equivalents—or bear a definite proportion to each other or, which is the same thing, if their magnitude differs according to the ||524| quantity of “labour” which they contain—then it is obvious that regarded as exchange-values, their substance must be the same.  Their substance is labour.  That is why they are “values”.  Their magnitude varies, according to whether they contain more or less of this substance.  But Ricardo does not examine the form—the peculiar characteristic of labour that creates exchange-value or manifests itself in exchange-values—the nature of this labour.  Hence lie does not grasp the connection of this labour with money or that it must assume the form of money.  Hence he completely fails to grasp the connection between the determination of the exchange-value of the commodity by labour-time and the fact that the development of commodities necessarily leads to the formation of money.  Hence his erroneous theory of money.  Right from the start he is only concerned with the magnitude of value, i.e., the fact that the magnitudes of the va1ues of the commodities are proportionate to the quantities of labour which are required for their production.  Ricardo proceeds from here and he expressly names Adam Smith as his starting-point (Chapter I, Section I).

Ricardo’s method is as follows: He begins with the determination of the magnitude of the value of the commodity by labour-time and then examines whether the other economic relations and categories contradict this determination of value or to what extent they modify it.  The historical justification of this method of procedure, its scientific necessity in the history of economics, are evident at first sight, but so is, at the same time, its scientific inadequacy.  This inadequacy not only shows itself in the method of presentation (in a formal sense) but leads to erroneous results because it omits some essential links and directly seeks to prove the congruity of the economic categories with one another.

Historically, this method of investigation was justified and necessary.  Political economy had achieved a certain comprehensiveness with Adam Smith; to a certain extent he had covered the whole of its territory, so that Say was able to summarise it all in one textbook, superficially but quite systematically.  The only investigations that were made in the period between Smith and Ricardo were ones of detail, on productive and unproductive labour, finance, theory of population, landed property and taxes.  Smith himself moves with great naïveté in a perpetual contradiction.  On the one hand he traces the intrinsic connection existing between economic categories or the obscure structure of the bourgeois economic system.  On the other, he simultaneously sets forth the connection as it appears in the phenomena of competition and thus as it presents itself to the unscientific observer just as to him who is actually involved and interested in the process of bourgeois production.  One of these conceptions fathoms the inner connection, the physiology, so to speak, of the bourgeois system, whereas the other takes the external phenomena of life, as they seem and appear and merely describes, catalogues, recounts and arranges them under formal definitions.  With Smith both these methods of approach not only merrily run alongside one another, but also intermingle and constantly contradict one another.  With him this is justifiable (with the exception of a few special investigations, [such as] that into money) since his task was indeed a twofold one.  On the one hand he attempted to penetrate the inner physiology of bourgeois society but on the other, he partly tried to describe its externally apparent forms of life for the first time, to show its relations as they appear outwardly and partly he had even to find a nomenclature and corresponding mental concepts for these phenomena, i.e., to reproduce them for the first time in the language and [in the] thought process.  The one task interests him as much as the other and since both proceed independently of one another, this results in completely contradictory ways of presentation: the one expresses the intrinsic connections more or less correctly, the other, with the same justification—and without any connection to the first method of approach—expresses the apparent connections without any internal relation.  Adam Smith’s successors, in so far as they do not represent the reaction against him of older and obsolete methods of approach, can pursue their particular investigations and observations undisturbedly and can always regard Adam Smith as their base, whether they follow the esoteric or the exoteric part of his work or whether, as is almost always the case, they jumble up the two.  But at last Ricardo steps in and calls to science: Halt!  The basis, the starting-point for the physiology of the bourgeois system—for the understanding of its internal organic coherence and life process—is the determination of value by labour-time.  Ricardo starts with this and forces science to get out of the rut, to render an account of the extent to which the other categories—the relations of production and commerce—evolved and described by it, correspond to or contradict this basis, this starting-point; to elucidate how far a science which in fact only reflects and reproduces the manifest forms of the process, and therefore also how far these manifestations themselves, correspond to the basis on which the inner coherence, the actual physiology of bourgeois society rests or the basis which forms its starting-point; and in general, to examine how matters stand with the contradiction between the apparent and the actual movement of the system.  This then is Ricardo’s great historical significance for science.  This is why the inane Say, Ricardo having cut the ground from right under his feet, gave vent to his anger in the phrase that “under the pretext of expanding it” (science) “it had been pushed into a vacuum”.  Closely bound up with this scientific merit is the fact that Ricardo exposes and describes the economic contradiction between the classes—as shown by the intrinsic relations—and that consequently political economy perceives, discovers the root of the historical struggle and development.  Carey (the passage to be looked up later) therefore denounces him as the father of communism.

I find the following, at least, of interest in this long passage:

• Marx here writes about "the connection as it appears in the phenomena of competition", "the external phenomena of life, as they seem and appear", "externally apparent forms of life". I think these phrases echo what Marx elsewhere describes as "vulgar political economy", commodity "fetishism", and the "illusions" created by competition.
• Marx criticizes Ricardo for only being concerned with "the magnitudes of values of commodities", not with the "peculiar character of labour that ... manifests itself in exchange values". I think this supports those who do not see a (great) contradiction between volumes 1 and 3 of Capital.
• Marx talks about the connection of labor values with money. I like interpretations or solutions of Marx's transformation problem that relate value to some abstract measure of the value of the output of a capitalist economy, namely:
  • Those based on Sraffa's standard commodity
  • Foley and Duménil's new interpretation, which focuses on the Monetary Expression of Labor Time (MELT).
• I quite like that "Halt!" I think it fair to say that Marx saw himself following and transcending Ricardo in exploring "the obscure structure of the bourgeois economic system", "the intrinsic relations", "the inner coherence, the actual physiology of bourgeois society".

Kalecki And Sraffa: Compatible?

Two Great Economists

1.0 Introduction

Michal Kalecki set out macroeconomic models in which markup pricing was common. Economists in this tradition rarely explore the effect of inter-industry flows on prices. Sraffians, on the other hand, usually specify prices, at least, to a first approximation, in a model of full competition. Can work in the traditions of Michal Kalecki and of Piero Sraffa be usefully combined?

2.0 A Model

Consider an economy in which n commodities are produced by n (single-product) industries. Inter-industry flows are described by a nxn matrix A, where a_{i, j} is the amount of the ith commodity used as input per unit output in the jth industry, at the given level of output of the jth industry. Labor inputs are described by the row vector a₀, where a_{0, j} is the quantity of labored hired in the jth industry per unit output, at the given level of output of the jth industry.

The positive constants m₁, m₂, ..., m_n represent barriers to entry among the different industries. The going rate of profits is earned in industries in which m_j is unity. Industries in which m_j exceeds unity have high barriers to entry. Perhaps a large scale of production is needed to operate profitably in such an industry. Industries with m_j less than unity are backwards, in some sense. At any rate, they earn less than the going rate of profits. These constants lie along the principal diagonal of the diagonal matrix M. That is, m_{i, j} is m_j, for i equal to j. And m_{i, j} is zero, for i unequal to j.

The row vector p represents prices, where p_j is the price of a unit quantity of the output of the jth industry. Suppose w represents the wage, and r represents The rate of profits.

The matrix A, the row vector a₀, the diagonal matrix M, and one of the distributive variables (say, the rate of profits r) are the given data for this model. The prices p and the remaining distributive variable (for example, wages w) are the unknowns to be found. One can set out the (modified) Sraffa equations for prices:

(p A M + a₀ w)(1 + r) = p

(I think models of full cost prices typically show markups being earned on both labor and material costs.) A numeraire should be specified. For example, one can set out the following normalization:

p₁ + p₂ + ... + p_n = 1

Likewise, the markups are only specified by the model, so far, up to a scalar multiple. I suggest the following normalization condition for markups:

m₁ x m₂ x ... x m_n = 1

Presumably this model can be extended, as in Sraffa (1960) to embrace fixed capital, land, joint production in general, and an analysis of the choice of technique.

3.0 Conclusion

The above has set out a model of prices of production. This model provides a framework for analyzing both the effects of inter-industry flows on prices and of markup pricing, arising from barriers to entry and other hindrances to full competition. The compatibility of some such model with both Kaleckian macroeconomics and the larger research agenda of Sraffa remains to be argued. Likewise, I have not shown the usefulness of this sort of model in empirical explanations of actual capitalist economies. One important issue in such discussions would probably be the applicability of models of prices of production to industries in which the planned operating level is less than full capacity.

This post should really have a bibliography, since the question of the compatibility of the economics of Kalecki and of Sraffa has been raised before. I gather that Paolo Sylos Labini, in some unpublished work in the 1960s, set out and analyzed a model rather like the above.

A System Block Diagram For Nicholas Kaldor

A System Block Diagram For A Business Cycle Model

I have previously presented a (replication of an) analysis of a discrete-time formalization of Kaldor's Keynesian model of business cycles. The system block diagram, above, is another way of specifying the model. This diagram, I think, helps make certain characteristics of the system more readily apparent:

• The non-linear component of the system, that is, the inverse tangent function, stands out.
• Only two state variables, national income (Y_t) and the value of the capital stock (K_t), need to be specified for this system.
• The ordered pair (Y_t, K_t) = (μ, σμ/δ) is a fixed point of the function specified by this system.

I am not sure about the use of one-step time lags to represent iteration for the Kaldor model. Presumably, Steve Keen has thought about this question for his software.

I think Keynes' General Theory can be read as leading towards systems thinking prior to its development in other disciplines.

Our Rulers Do Not Know Why They Dislike Government Debt

Table 3: The Perceived Importance of Problems Facing U.S.A.

Problem	% Wealthy Saying "Very Important"
Budget deficits	87
Unemployment	84
Education	79
International terrorism	74
Energy supply	70
Health care	57
Child poverty	56
Loss of traditional values	52
Trade deficits	36
Inflation	26
Climate change	16

A few weeks ago, Paul Krugman mentioned a recent paper by Page, Bartels, and Seawright. I believe it is this one:

• Benjamin I. Page, Larry M. Bartels, and Jason Seawright (March 2013). Democracy and the Policy Preferences of Wealthy Americans, Perspectives on Politics, V. 11, No. 1: pp. 51-73.

This paper reports a pilot study on the political views of the wealthiest Americans. The authors gathered data in interviews with residents drawn from a sample of the very wealthy in Chicago. Page et al. motivate their interest in the policy preferences of wealthy Americans by noting recent research demonstrating that the vast majority of the country has little to no influence on policy decisions made in the Federal government. They hope to expand their research to a national sample in the future.

They report views on many areas of public policy. Generally, our rulers are reactionary and the opposite of benevolent. Business backgrounds in finance or industry, inherited wealth or "earned" wealth, were not correlated with differences in views. The sample size might be too small to provide enough power to distinguish, among the wealthy, effects of where they sit on where they stand. Professionals, mainly lawyers and doctors, tended to be slightly less reactionary.

Above, I reproduce Table 3 from this paper. Those surveyed "think" government budget deficits are the biggest problem facing the United States. One might suggest that lowering such deficits could be only an intermediate, instrumental goal. But towards what end? Page et al. note that they do not seem worried about deficits leading to high rates of inflation; notice how low inflation is as a worry. Page et al. suggest that the wealthy have bought into the "crowding out" argument. Of course, theoretically, supply and demand for savings does not determine interest rates. Empirically, the crowding out argument makes no sense in the current conjuncture either.

I have an old explanation of this puzzle. Paul Krugman recently cited Michal Kalecki's explanation of why capitalists dislike increased government spending in depressions, even though such fiscal policy successfully dampens downswings in business activity. Krugman is not just depending on the capability of Kalecki's explanation to make sense of history long post-dating Kalecki's contribution. Krugman is also aware of the quantitative survey data I cite above.

Planning Empirically Superior To Markets: The Fixed Microwave Spectrum

This post notes the existence of the following article:

• Mitchell Lazarus (May 2013). When Spectrum Auctions Fail, IEEE Spectrum: pp. 33-35, 56-59.

This article is about the microwave spectrum, in the range from 3 to 100 Gigahertz, with an emphasis on the commercial use of the low end of this range. From World War II until fairly recently, conflicts and potential interference in the use of the microwave spectrum were resolved by discussions among engineers working for the users of the conflicting resources. Nowadays, conflicts are resolved by auctioning off the spectrum. (Presumably, these auctions are inspired by the work for which the so-called Nobel prize in economics was awarded last year.) And, Lazarus argues, these auctions have failed to work as well as the previous regime did.

Lazarus provides a popular survey of some technical characteristics of microwave radiation. Microwave is used for point-to-point communication, not for broadcast. This use often parallels a physical infrastructure in an area. The auctions typically leave the frequencies put up for auction underused, or so Lazarus argues.

A Near-Term Goal

I would like to develop a numeric example with:

• Smooth production functions, and
• Properties analogous to the ones highlighted in this example.

One of the parameters of the utility functions in this example expresses the willingness of consumers to defer consumption. A greater willingness to defer consumption supposedly represents a greater supply of "capital", in some sense. In a "perverse" case, this greater supply, all else the same, is associated with a long run equilibrium with a higher equilibrium interest rate.

I do not think that the "perversity" I am trying to illustrate depends on the distinction between discrete technologies and smooth production functions. I am aware, however, of a theorem that applies to a technology with smooth production functions, but not to discrete technology:

Theorem: Consider an economy in which all produced commodities are basic, in the sense of Sraffa, for all feasible techniques. And suppose the production of one commodity can be described by a continuously differentiable production function. Then this economy cannot exhibit reswitching of techniques.

The relevance of this theorem to my goal is not clear. I am willing to consider examples with non-basic goods. So examples should be possible to construct with smooth production functions and reswitching. But I do not even need reswitching. I am merely looking for capital-reversing. And I do not even insist that real Wicksell effects be positive. I will be content with positive price-Wicksell effects swamping negative real Wicksell effects.

Maybe the kind of example I am seeking is set out in a end-of-the-chapter problem in Heinz D. Kurz and Neri Salvadori's 1997 book, Theory of Production: A Long-Period Analysis (Cambridge University Press).

By looking at the convexity of the wage-rate of profits curves on the frontier, one can read off the direction of price Wicksell effects. And I have already shown that an example can be created with Cobb-Douglas production functions and positive price Wicksell effects. I have yet to examine the relative sizes of price and real Wicksell effects in the example, derive conditions on their directions and sizes, or create a numeric example satisfying those conditions.

Eventually, I would like to explore the dynamics of non-stationary equilibrium paths in such a model built on unarguably neoclassical premises. The point is to continue an internal critique of neoclassical microeconomics, not to describe actually existing capitalist economies.

Suggestions For Adding To The Stack

I probably will not order the first two. But I think their existence is of interest. And I do not currently have access to the third.

• Norbert Häring and Niall Douglas (2012) Economists and the Powerful: Convenient Theories, Distorted Facts, Ample Rewards, Anthem Press.
• Kalle Lasn (2013) Meme Wars: The Creative Destruction of Neoclassical Economics, Seven Stories Press.
• Tobias Galla and J. Doyne Farmer (22 January 2013). Complex Dynamics in Learning Complicated Games, Proceedings of the National Academy of Sciences of the United States of America, V. 110, No. 4: pp. 1232-1236
• Sergio Parrinello (2000). The "Institutional Factor" in the Theory of International Trade: New vs. Old Trade Theories.

I suppose I might try to find the paper, by Benjamin Page, Larry Bartels, and Jason Seawright, that Paul Krugman references in his New York Times column last Friday. By the way, Krugman is basically worrying that economics is "vulgar political economy", a technical term introduced by Karl Marx. But Krugman cannot reference Marx or acknowledge Marx was maybe correct about something.

In my draft paper on the failure of the theory of comparative advantage to justify free trade, I am currently ignoring Krugman and new trade theory. The fourth reference above might be usefully footnoted in my article. I believe Parrinello also has an article in a recent festschrift volume for Ian Steedman.

I recently stumbled across Rob Beamish's 1992 book, Marx, Method, and the Division of Labor. This book traces the development of a concept, the division of labor, in Marx's manuscripts and published work, including the manuscripts I mentioned in a previous post. Furthermore, Beamish argues that if historical materialism is true, it must apply to the development of Marx's ideas.

Choice of Technique, A Two Good Model, Cobb-Douglas Production Functions

Figure 1: Wage-Rate of Profits Curves and their Frontier

1.0 Introduction

This post is a generalization of a neoclassical one-good model. It advances a comparison of Sraffian analysis of the choice of the cost-minimizing choice of the technique and neoclassical analyses, correctly understood, of marginal productivity. Accordingly, all production functions are smooth in this example. If substitutability is seen as a technological property of production functions, then the single capital good and labor can be substituted in each of the two industries in this model.

2.0 The Technology

Consider a simple economy in which steel and corn are produced from inputs of steel and labor. The steel used as an input in production is totally used up in yearly cycles, and the outputs become available at the end of the year. In other words, this is a model without fixed capital, and all production processes require a year to complete.

2.1 Production Functions

The production function for steel is:

Q₁ = F₁(X₁, L₁) = A₁ X₁^α₁ L₁^{(1 - α₁)}

where:

• Q₁ is (gross) output of steel (in tons).
• X₁ is steel (tons) used as a capital good in the steel industry.
• L₁ is labor (person-years) used as an input in the steel industry.

and A₁ and α₁ are positive constants such that:

0 < α₁ < 1

The production function for corn is:

Q₂ = F₂(X₂, L₂) = A₂ X₂^α₂ L₂^{(1 - α₂)}

where:

• Q₂ is (gross) output of corn (in bushels).
• X₂ is steel (tons) used as a capital good in the corn industry.
• L₂ is labor (person-years) used as an input in the corn industry.

and A₂ and α₂ are positive constants such that:

0 < α₂ < 1

2.2 A Set of Coefficients of Production

An alternative specification of this Constant-Returns-to-Scale (CRS) technology is as a set of coefficients of production a₀₁(s₁), a₀₂(s₂), a₁₁(s₁), a₁₂(s₂) from the set:

{ (a₀₁(s₁), a₀₂(s₂), a₁₁(s₁), a₁₂(s₂)) | 0 < s₁, 0 < s₂}

where:

a₀₁(s₁) = [1/(A₁s₁)]^{[1/(1 - α₁)]}

a₀₂(s₂) = [1/(A₂s₂)]^{[1/(1 - α₂)]}

a₁₁(s₁) = s₁^(1/α₁)

a₁₂(s₂) = s₂^(1/α₂)

and

• a₀₁(s₁) is the labor required, in the steel industry, per ton steel produced.
• a₀₂(s₂) is the labor required, in the corn industry, per bushel corn produced produced.
• a₁₁(s₁) is the steel input required, in the steel industry. per ton steel produced (gross).
• a₁₂(s₂) is the steel input required, in the corn industry, per bushel corn produced.

2.0 Quantity and Price Equations, Given the Technique

Consider a stationary state in which the firms employ one person-year of labor each year, and prices are stationary. For notational convenience below, define the following function:

f(R) = (a₀₁a₁₂ - a₀₂a₁₁)R + a₀₂

2.1 Quantity Relations

The amount of steel produced each year, measured in tons, is:

q₁ = a₁₂/f(1)

The amount of corn produced each year, measured in bushels, is:

q₂ = (1 - a₁₁)/f(1)

These quantities must satisfy two equalities. First, the amount of labor employed is unity:

1 = a₀₁q₁ + a₀₂q₂

Second, consider the following equation:

q₁ = a₁₁q₁ + a₁₂q₂

The left-hand side of the above equation denotes the quantity of steel produced each year and available, as output from the steel industry, at the end of each year. The right-hand side denotes the sum of steel used as inputs in the steel and corn industries, respectively. These inputs must be available at the start of each year. Hence, the above equation is a necessary condition when the economy is in a self-sustaining, stationary state.

2.2 Price Relations

I take the consumption good, corn, as the numeraire. The price of steel, in units of bushels per ton, is

p = a₀₁/f(1 + r),

where r is the rate of profits. The wage is:

w = [1 - a₁₁(1 + r)]/f(1 + r)

The above equation is known as the wage-rate of profits curve.

The price of steel, the wage, and the rate of profits must satisfy two equations. The condition that the price of steel just cover the cost of producing steel is:

pa₁₁(1 + r) + a₀₁w = p

The left-hand side of the above equation shows the cost of producing a ton of steel. Costs are inclusive of normal profits, so to speak, on the cost advanced to purchase physical inputs at the start of the year. In this case, those inputs consist of steel, the single capital good in this model. Although labor is hired at the start of the year to work throughout the year, the price equations in this model show labor being paid out of the harvest gathered at the end of the year.

The condition that the price of corn just cover the cost of producing corn yields a similar equation:

pa₁₂(1 + r) + a₀₂w = 1

2.3 The Capital-Labor Ratio

"Capital" is an ambiguous term. It denotes both physically-existing means of production. And it denotes the value of those means of production, when embedded in certain social relations. For example, in this model, the distribution of the capital goods over the two industries is assumed to be appropriate to the continued self-reproduction of the economy. In a sense, the plans of entrepreneurs and firms managers are coordinated.

At any rate, the relationships described so far allow one to express the value of capital, in numeraire units, per person-years, given the technique:

k = p q₁

k = a₀₁a₁₂/[f(1)f(1 + r)]

The capital-labor ratio (in units of bushels per person-years) does not appear in any legitimate marginal product. Nevertheless, I find it a useful quantity for further analysis in multicommodity models.

3.0 The Chosen Technique

The cost-minimizing technique differs with the rate of profits. For analytical convenience, I take the rate of profits as exogenous in this model. One could, instead, if one so chose, take the wage as given and find the rate of profits endogenously. At any rate, this model is open, and the distribution of income is not determined in the model. The equations below set out each of the four coefficients of production in this model as functions of the rate of profits:

a₀₁ = (1/A₁)^{[1/(1 - α₁)]} [(1 + r)/α₁]^{[α₁/(1 - α₁)]}

a₀₂ = (1/A₂)

x {(1 - α₂)/[(α₁)^{[α₁/(1 - α₁)]}(1 - α₁)α₂]}^α₂

x [(1 + r)/A₁]^{[α₂/(1 - α₁)]}

a₁₁ = α₁/(1 + r)

a₁₂ = (1/A₂)

x [(α₁)^{[α₁/(1 - α₁)]}(1 - α₁)α₂/(1 - α₂)]^{(1 - α₂)}

x [A₁/(1 + r)]^{(1 - α₂)/(1 - α₁)}

3.1 Steel as a Basic Commodity and the One-Good Case

I have previously set out an analysis of the choice of technique for a one-good model with an aggregate Cobb-Douglas production function. In the two-good model set out in this post, the coefficients of production for steel, a₀₁ and a₁₁, when the cost-minimizing technique is chosen, are the same as the coefficients of production in that one-good model. This is not surprising.

In the model in this post, steel enters, as an input, into the production of both steel and corn, for all possible techniques. On the other hand, corn never enters as an input into the production of any commodity. In the technical terminology of post-Sraffian economics, steel is always a basic commodity, and corn is never a basic commodity. Thus, the production of steel can be analyzed, in some sense, prior to the analysis of the production of corn.

3.2 A One-Good Special Case

Consider the special case in which:

α₁ = α₂ = α

A₁ = A₂ = A

In effect, steel and corn are the same commodity. The coefficients of production, for the cost-minimizing technique are:

a₀₂ = a₀₁ = (1/A)^{[1/(1 - α)]} [(1 + r)/α]^{[α/(1 - α)]}

a₁₂ = a₁₁ = α/(1 + r)

So this case reduces to the one-good model, as it should. This concludes my analysis of this special case.

4.0 The Chosen Technique on Unit Isoquants and Marginal Productivity Conditions

The coefficients of production are such that the steel industry lies on its unit isoquant:

1 = F₁(a₁₁, a₀₁)

Likewise, the corn industry lies on its unit isoquant:

1 = F₂(a₁₂, a₀₂)

Since the coefficients of production in Section 3 above are for the cost-minimizing technique, all valid marginal productivity relationships must hold. I have chosen to express each marginal productivity condition in numeraire units per unit input. And, the cost of an input and its marginal product are equated here at the end of the year.

Following these conventions, the following display equates the cost of steel to the value of the marginal product of steel in the steel industry:

p(1 + r) = p ∂F₁(a₁₁, a₀₁)/∂a₁₁

Likewise, the following display equates the cost of steel to the value of the marginal product of steel in the corn industry:

p(1 + r) = ∂F₂(a₁₂, a₀₂)/∂a₁₂

Since wages are paid out of the harvest, the rate of profits does not appear in my statement of marginal productivity conditions for labor. The following display equates the wage and the value of the marginal product of labor in the steel industry:

w = p ∂F₁(a₁₁, a₀₁)/∂a₀₁

Likewise, the following display equates the wage and the value of the marginal product of labor in the corn industry:

w = ∂F₂(a₁₂, a₀₂)/∂a₀₂

I have checked the above equations for the isoquants and the four marginal productivity equations. This is quite tedious.

Above, I have listed six equations, two expressing the condition that the coefficients of production lie upon unit isoquants and four marginal productivity equations. These six equations are sufficient to determine the six unknowns (w, p, a₀₁, a₀₂, a₁₁, and a₁₂) in terms of the model parameters and the externally specified rate of profits. In other words, this model illustrates that marginal productivity is a theory of the choice of technique, not of the (functional) distribution of income.

5.0 The Wage-Rate of Profits Frontier

An alternate analysis of the choice of technique can be based on the wage-rate of profits frontier. And this analysis yields the same answer as the above analysis based on marginal productivity.

Recall, from Section 2.2, that a technique can be specified as an ordered pair chosen from the specified index set. The index variables for the cost-minimizing technique, as a function of the rate of profits are:

s₁ = [α₁/(1 + r)]^α₁

s₂ = (1/A₂)^α₂

x [(α₁)^{[α₁/(1 - α₁)]}(1 - α₁)α₂/(1 - α₂)]^{[(1 - α₂)α₂]}

x [A₁/(1 + r)]^{[(1 - α₂)α₂/(1 - α₁)]}

I think it of interest to note that both the optimal process for producing steel and the optimal process for producing corn, in a stationary state, vary continuously with the rate of profits. This is not a generic result for a discrete technology. In a discrete technology, the cost-minimizing techniques at a switch point typically differ in the process used in only one industry; a small variation in the rate of profits thus affects only the specification of a process in one industry.

5.1 First Order Conditions

Since the coefficients of production are functions of the index variables, the wage-rate of profits curve for a technique can be viewed as a function of:

• The index variables s₁ and s₂,
• The rate of profits r, and
• The model parameters α₁, A₁, α₂, and A₂.

A necessary condition for a technique to be cost-minimizing, at a given rate of profits, is that the wage be a maximum. This maximum is taken from the wage on each wage-rate of profits curve, over all techniques. In the current context, with a model with smooth production functions, the first derivative of the wage-rate of profits frontier, with respect to each index variable, must be zero at the maximum:

∂w/∂s₁ = 0

∂w/∂s₂ = 0

Note that the above is a system of two equations in the two unknown index variables. I did not actually calculate the above derivatives for this model. Perhaps Figure 1 provides some confidence in this mathematics. I deliberate drew three wage-rates of profits curves on the frontier and one off of it.

5.2 Second Order Conditions

The FOCs determine a critical point. The calculus is consistent with such a critical point being a local maximum, a local minimum, or a saddle point. The following are sufficient conditions, in this context, for a critical point to be a local maximum:

∂²w/∂s₁² < 0

∂²w/∂s₂² < 0

D(s₁, s₁) > 0

where D(s₁, s₁) is defined by:

D(s₁, s₁) = [∂²w/∂s₁²][∂²w/∂s₂²] - [∂²w/∂s₁∂s₂]²

Of the three SOCs, either the first or the second is redundant.

6.0 Conclusion

I still have some ideas for future work with this model. But I think this is enough for one blog post. I hope the above presentation suggests that marginal productivity is not a theory of distribution, in general. One cannot validly hold, for example, that real wages are determined by the marginal product of labor. Furthermore, the Sraffian analysis of the choice of technique is analytically equivalent to the determination of the choice of technique, given, for example, the rate of profits, by marginal productivity.

Who Is Joshua Clover?

Joshua Clover has a great one-page article on Krugman in this week's Nation. I'd like to quote the whole thing. But I'll make do with extracts:

"Consider the phenomenon of Paul Krugman, of late taking a curious turn... ...Krugman [is] advantageous[ly] position[ed] as a public intellectual famously handy with hard data and rigorous analyses. Ask Thomas Friedman: anyone can be a blowhard on matters global. Few can do the math.

...as a star economist, [Krugman's] historical role has been to reinvigorate the duel between liberal Keynesians and the recently regnant monetarists of various stripes...

...For the record, I greatly preferred the Backstreet Boys to 'N Sync...

The oppositions Republican/Democrat and monetarist/Keynesian are in this regard pure pop. They are, as you will have noticed some time ago, choices only in the most straitened sense: minimally distinct management strategies for capitalism. Their present distinction lies in whether crisis is best managed by allowing the owners of capital everything they want immediately, or at pace lest they choke on something...

And yet. In December, Krugman wrote two blog entries in swift succession: 'Rise of the Robots' and 'Human Versus Physical Capital'. Inequality, his charts informed him, was itself a consequence of the opposition between capital and labor—specifically the increasing domination of capital in the form of machines—as labor is expelled from the production process. That ratio turns out to be basically the same measure as productivity, sine qua non of economic progress.

Moreover, in a development Krugman couldn't quite bring himself to declare, his charts suggest that a generally declining labor share since the 1970s has also spelled bad news for overall profitability outside the finance sector. The productivity race wasn't just unfortunate for the unemployed; it was for capital a poison pill of its own making. Thus Krugman's comedy: always on the verge of discovering the arguments of a 150-year-old book; always turning away at the last second. In Krugman's words, 'I think our eyes have been averted from the capital/labor dimension of inequality, for several reasons. It didn't seem crucial back in the 1990s, and not enough people (me included!) have looked up to notice that things have changed. It has echoes of old-fashioned Marxism—which shouldn't be a reason to ignore facts, but too often is. And it has really uncomfortable implications.'

Does it? I suppose so. And that uncomfort is what pop, for all its pleasures, must defer. Pop must affirm the way things are, no matter how often it choruses the word 'change.' You cannot be Paul Krugman, Pop Star, and at the same time discover that capital is built to break us, and itself—even if your charts so testify. So you will not be shocked to discover Krugman stepped back from this realization and continued about his business, scarcely speaking of it again. There are some things you do not say. They are not popular."

"Technology and Wages, the Analytics" was another Krugman post in the same period, along the same lines. I've already commented on that one.

Choice Of Technique With A Smooth Aggregate Production Function

Figure 1: Coefficients of Production for the Technology

1.0 Introduction

This post advances, somewhat, my start at a reconsideration of the dynamics of Overlapping Generations Models (OLGs). Only the production side of a stationary state is considered here. Furthermore, only a very special case - namely, a one-good model - is analyzed here.

I guess the most exciting aspect of this post is an illustration of the claim that the construction of the wage-rate of frontier is useful for the analysis of the choice of technique for "smooth" production functions, not just for discrete technologies. I have never understood, for at least a quarter of a century, why some economists seem to talk as if a fundamental distinction exists between such models. In some contexts, some conclusions differ. But it seems to me to be silly to say that the Cambridge Capital Controversy turns around an empirical question on the degree of substitutability of inputs in production.

2.0 Specification of Technology

Consider a simple economy in which corn is produced from inputs of labor and corn. Assume the existence of Constant Returns to Scale (CRS). A technique is specified by an ordered pair of coefficients of production, where each ordered pair is from a set containing a continuum of such ordered pairs:

{ [a₀(s), a₁(s)] | 0 < s < 1}

where:

a₀(s) = 1/(A s)^{1/(1 - α)}

a₁(s) = s^1/α

and α and A are specified positive parameters such that:

0 < α < 1

Figure 1 graphs the coefficients of production as a function of the index s. All graphs are draw for a value of α of 1/4 and of A of 5.

3.0 Derivation of the Cobb-Douglas Production Function

The above specification of the technology shows, for a unit output of corn, a smooth trade-off of inputs of labor and corn inputs. This specification of technology allows for the derivation of a conventional production function. The following is an equation for a unit isoquant for this technology:

1 = A [a₁(s)]^α [a₀(s)]^{1 - α}

Define:

• Q is (gross) corn (bushels) output.
• L is labor (person-years) input.
• X is (seed) corn input.

From CRS, it follows:

Q = A [Q a₁(s)]^α [Q a₀(s)]^{1 - α}

Or:

Q = A X^α L^{1 - α}

The last equation above is how the (in)famous Cobb-Douglas production function is typically represented. So the specification of technology used in this post is a (non-unique) representation of a Cobb-Douglas production function.

4.0 Analysis of the Choice of Technique

For a given technique, Sraffa's price equations become one equation:

a₁(s)(1 + r) + a₀(s) w = 1

where:

• r is the rate of profits
• w is the yearly wage (in units of bushels per person-year).

The price equation embeds the assumptions that production of corn requires a year to complete and that labor is paid out of the yearly harvest. One can derive a wage-rate of profits curve from the price equations:

w(r, s) = [1 - a₁(s)(1 + r)]/a₀(s)

In this case, each wage-rate of profits curve is a straight line. Figure 2 shows three selected wage-rate of profits curves.

Figure 2: Wage-Rate of Profits Curves and Their Frontier

Figure 2 shows, in violet, the outer wage-rate of profits frontier. When firms choose the cost-minimizing technique in a steady state, the economy will lie on this curve. (In this case, with a continuum of techniques, each point on the frontier is a non-switch point.) A closed-form expression for the wage-rate of profits frontier is easily derived. The First Order Condition (FOC) for the choice of technique can be expressed as equating the derivative, with respect to the index variable, of the wage-rate of profits curve to zero:

dw/ds = 0

The FOC yields an equation which can be solved for the index variable:

s(r) = [α/(1 + r)]^α

So the coefficients of production, for the cost-minimizing technique, can be found as functions (Figure 3) of the rate of profits:

a₀(r) = [1/A^{1/(1 - α)}] [(1 + r)/α]^{α/(1 - α)}

a₁(r) = α/(1 + r)

Thus, the desired expression for the wage-rate of profits frontier is:

w(r) = (1 - α) A^{1/(1 - α)} [α/(1 + r)]^{α/(1 - α)}

In this special case, the desired amount of labor per unit output is higher, the lower the wage. Likewise, the desired amount of the capital good per unit output is lower, the higher the rate of profits. These results do not generalize to multi-commodity models.

Figure 3: Optimal Coefficients of Production

5.0 Capital Intensity

In this special case, the ratio of the value of capital goods to labor can be calculated in physical terms, without addressing a question of valuation. That is, the capital-labor ratio, as a function of the rate of profits (Figure 4), is easily derived:

I(r) = a₁(s(r))/a₀(s(r)) = [α A/(1 + r)]^{1/(1 - α)}

In this special case, the capital-labor ratio is a downward-sloping, single-vauled function of the rate of profits. These properties do not generalize, either.

Figure 4: Capital Intensity

"Economics Textbooks - Decades of Scientific Fraud"

Lars Syll has written a post, titled "Economics Textbooks - Decades of Scientific Fraud". If you had not already read it, could you guess what it is about from the title?

I would expect likely guesses to be non-unique. It is not about:

• How the Cambridge Capital Controversy demonstrates that textbook teaching on labor markets and, for example,on the minimum wage is nonsense.
• The incoherence of textbook teaching on the justification for lack of tariffs by the theory of comparative advantage.
• The textbook misrepresentations of the theories of various economists, including John Maynard Keynes.

You can extend the above list at your leisure.

In many ways, economics seems to me to be an extraordinary subject. Good arguments have existed for decades for discarding most of mainstream teaching and practice. As far as I can see, the bulk of these arguments, including their very existence, are just ignored by most mainstream economists. I am willing to entertain demonstrations of the fallacy of theories taught in almost all mainstream textbooks for decades. I think my willingness to explore other demonstrations than those I have been previously aware of is partly due to my belief that most economists are socialized into willful ignorance.

I can see why some young mainstream economists may resist the notion that they have been taught, mostly, lies and nonsense. And so they may look in the research literature for arguments against the arguments and demonstrations that I accept, or at least try to explore. Since my favorite positions were established after long controversy, you can find neoclassical counter-arguments, of a sort. For example, one might cite, in response to the Cambridge Capital Controversy:

• Edwin Burmeister's championing of Champernowne's chain index measure of capital¹.
• Frank Hahn's advocacy, including in response to the Cambridge Capital Controversy, of General Equilibrium Theory².

In response to the application of the CCC to the theory of international trade, one might cite:

• Christopher Bliss's suggestion that the necessary existence of gains from trade follows from including an assumption that all produced goods, not just consumer goods, be traded internationally.
• Wilfred Ethier's claim that the endowment of capital be calculated in equilibrium prices in models of international trade.
• A suggestion that the theory of international trade be organized around comparisons of intertemporal equilibrium paths³.

One might think a conclusion is more justified from the weight of the evidence when multiple arguments reach that conclusion. So one might react to existence of such controversies in the research literature as allowing one to support mainstream teaching. However, one would be wrong in this attitude. If you look at these responses in some detail, you will find that the orthodox economists do not end up at the textbook position, but at some other point. But, as far as I can tell, neither side ends up being transitioned from the research literature to conventional teaching. Would you not be more confident in adopting some such conventional counter-argument to one of my favorite arguments if it were widely taught? Otherwise, should you not suspect yourself of adopting an idiosyncratic misinterpretation of the theory?

Footnotes

1. This chain index is endogenous, not exogenous, as needed for much of neoclassical theory. Furthermore, Burmeister accepts the validity of demonstrations of reswitching and capital-reversing.
2. A focus on intertemporal and temporary equilibria is a rather drastic change of theory from the traditional neoclassical focus on a comparison of long run equilibria. The latter comparisons seem to be to provide the (exploded) foundation for most mainstream policy advice.
3. Avinash Dixit (May 1981). The Export of Capital Theory, Journal of International Economics. V. 11, Iss. 2: pp. 279-294.

Perfect Competition Is The Same As Monopoly If You Do The Math Right

1.0 Introduction

This post summarizes one aspect of a theorem presented and proved by Roy Radner (1980). I have previously expressed skepticism about the claim in the post title. I have also heard that, in game theory, anything can happen, but nothing need happen. So, I suppose, one should not be surprised in stumbling over a proof of the existence of almost any market behavior in the literature on game theory. But I was surprised.

2.0 Selected Assumptions

2.1 Non-Cooperative Firms

In the model considered here, no mechanism exists to enforce agreements among firms. In the jargon, only (extensions of) Cournot-Nash equilibria are considered here.

2.2 Firm Managers Making Approximately Optimal Output Decisions

Although not commonly stated, the textbook presentation of perfect competition assumes the managers of the firms are systematically mistaken about their optimum decisions. A homogeneous product is assumed to be produced by a finite number of firms in the industry, and the total industry output is finite. Managers are assumed to disregard any strategic reaction by other firms to variation in their own firm's output and to take the price of their product as given. But, for a given consumer demand function, the firm's (notional) variation in output results in a variation in prices. So the decisions of the managers can only be approximately optimal, in textbook theory.

Radner proposes the notion of an epsilon-equilibrium to formalize this idea that firm strategies are only approximately optimal. In such an equilibrium each firm's strategy is such that, for example, average profit is within epsilon of the maximum average profit achieved by an optimal strategy, given the strategies of all other firms. As is typical in mathematical analysis, one should think of ε as a given (small) parameter.

2.3 Sequential Market Interactions

Firms are not considered as deciding on a single quantity to produce in this model. Rather, each firm decides on a sequence of T quantity outputs, one for each of T successive periods. The parameter T is known as the lifetime of the industry. Each firm decides on the output in a given period as a function of the outputs of all firms in all previous periods. A strategy is a sequence of such functions, one for each firm. The firm chooses a strategy to maximize its average or total discounted profit over the lifetime of the industry.

2.4 Replication

The theorem outlined here is used to compare epsilon-equilibria for different (finite) numbers of firms in the industry. Radner defines the replication case to apply when the demand price is an unchanged function of the average output per firm. In some sense, the number of consumers increases, in the model, with the number of firms.

3.0 An Informally Stated Theorem

Theorem: Consider the model with the above assumptions. Let the number of firms increase, along with the lifetime of the industry, such that the number of firms remains small enough, when compared to the lifetime of the industry. For any finite number of firms, equilibria exist in which the firms act as a cartel, and the cartel lasts for any given duration, provided the lifetime of the industry is taken large enough.

4.0 Conclusion

I think of the point of this post to explore the result of tweaking textbook assumptions in the theory of perfect competition. Apparently, the results are sensitive to the exact statement and combination of assumptions. I gather that further research in microeconomic theory has confirmed that whether or not equilibria converge, as the number of firms increase, to the perfect competition model is a fine point. That is, equilibria may or may not converge to a model with a continuum of firms. Radner seems to feel exploring certain sets of assumptions is of more interest than other sets. I have chosen to emphasize a set of assumptions in which any finite number of firms may act like a monopoly, in a precise sense.

Another approach might be better in empirically describing firms in actually-existing capitalism.

Reference

• Roy Radner (1980). Collusive Behavior in Noncooperative Epsilon-Equilibria of Oligopolies with Long but Finite Lives, Journal of Economic Theory, V. 22: pp. 136-154

A Numeric Example Of The Loss From Trade

Ratio of the Value of Capital to Labor

This post summarizes a numeric example in which at least one country is unambiguously worse off under free trade. This example illustrates the model I developed in a draft paper. The example in this post differs from the one in my paper; interest rates of concern here are more reasonable values.

The model is of two small open economies facing identical technologies for producing two consumer goods. The model assumptions are:

1. Two countries, A and B, can produce the same two commodities, wine and silk, for consumption.
2. The entrepreneurs in each country know the given flow-input, point-output technology (Table 1). Wine and silk each require two years of labor input per unit output. For example, the grapes for a unit of wine require 10 person-years of unassisted labor to be expended in the first year. One hundred eighty eight person-years of labor work up these grapes into wine produced for consumption at the end of the second year.
3. Each country has a given endowment of labor, the only non-produced factor of production in each country. The labor force is fully employed in each country.
4. Only commodities produced for consumption can be traded internationally. Laborers neither immigrate nor emigrate. Capital cannot be traded internationally.
5. Wine and silk are produced with different factor-intensities, silk being more capital-intensive and wine being more labor-intensive. No factor-intensity reversals exist.
6. All consumers, in all countries, have identical homothetic utility functions.
7. Perfect competition obtains in all markets; transport costs are negligible; and free trade exists in all commodities produced for consumption, unless otherwise specified.

These are textbook assumptions. The numeric example proves mainstream textbooks are simply incorrect, since the opposite answer is obtained.

Table 1: The Technology

	Country A	Country B
Wine Production	l1,A = 10 person-yrs per unit wine	l1,B = 10 person-yrs per unit wine
	l2,A = 188 person-yrs per unit wine	l2,B = 188 person-yrs per unit wine
Silk Production	l3,A = 100 person-yrs per unit silk	l3,B = 100 person-yrs per unit silk
	l4,A = 89 person-yrs per unit silk	l4,B = 89 person-yrs per unit silk
Endowments	lTotal,A = 4,158 person-years	lTotal,B = 3,969 person-years

The firms in both countries face given prices of wine and silk on the international market, as shown in Table 2. The domestic interest rate and the corresponding wage vary between the two countries. As shown in my paper, one can use this price system to determine which commodity, if any, firms in each country would find it most profitable to specialize in the production of. If the interest rate were zero, each country would attempt to specialize in the producing silk. For the price of silk to be a switching price, where firms would find it profitable to specialize in the production of both wine and silk, the interest rate must be 10% for the example. For the interest rates shown, country A specializes in the production of wine, and country B specializes in the production of silk.

Table 2: The Selected Price System

	Country A	Country B
Price of Silk:	p = 1 units wine per unit silk
Interest Rate:	rA = 20%	rB = 5%
Wage:	wA = (1/200) units wine per person-yr	wB = (1/194) units wine per person-yr

Given the technology, endowments, an equilibrium price system, and tastes, one can calculate how much wine and silk will be produced and consumed in each country, both when neither country can trade consumer goods on international markets and when both can. Table 3 shows the resulting patterns of consumption among stationary states in the two countries. The consumers in country A are unambiguously worse off in a stationary state with specialization and free trade.

Table 3: Results for the Numeric Example

		Autarky	Specialization
Wine Consumption	Country A	10 1/2 Units wine	10 1/2 Units wine
	Country B	10 1/44 Units wine	10 1/2 Units wine
	Total	20 23/44 Units wine	22 Units wine
Silk Consumption	Country A	11 Units silk	10 1/2 Units silk
	Country B	10 1/2 Units silk	10 1/2 Units silk
	Total	21 1/2 Units silk	22 Units silk

Figure 1, constructed for the example, shows that the endowment of capital cannot be taken as a parameter in the illustrated model. Because of price Wicksell effects, the quantity of capital varies with the interest rate, even for a given pattern of specialization. Yet confused textbook writers often present the Heckscher-Ohlin-Samuelson model in a two-country, two-commodity, two-factor framework, with the factors of production incorrectly labeled as "labor" and "capital".

So much for the orthodox theory of free trade. Neo-Ricardians proved, more than a third of century ago, that the neoclassical theory of international trade is defective in other ways, too.

Political Elites Bowing Down Before The Ones They Serve

Table 1: Politicians in State Legislatures Ignorant of Strength of Constituent Support for Universal Health Care

Table 2: Politicians in State Legislatures Ignorant of Strength of Constituent Support for Gay Marriage

In 2012, Broockman and Skovron surveyed candidates for office in state legislatures throughout the United States. Nearly 2,000 candidates replied. About half of those responding won their races, about half are Democrats, and about half are Republicans. The survey asked the respondents to estimate their constituents' support for the following three policy proposals:

• Implement a universal healthcare program to guarantee coverage to all Americans, regardless of income.
• Same sex couples should be allowed to marry.
• Abolish all federal welfare programs.

Broockman and Skovron also estimated the actual support for these proposals in each of the respondents' districts. Estimates of actual support come out of a multi-level regression and poststratification (MRP) model. The paper contains a neat map of greater Los Angeles showing the results of the MRP model for districts there.

Figures 1 and 2, above, compare the actual support for the first two policy proposals, respectively, to estimated support. If estimates matched actual values, they would lie on the 45 degree line, shown in grey on the graphs. A striking finding is that members of state legislatures tend to think their districts are more conservative than they are. The bias is more extreme for conservative politicians: "Nearly half of sitting conservative officeholders appear to believe that they represent a district that is more conservative ... than the most conservative legislative district in the entire country." Furthermore, politicians learn next to nothing about their constituents' views in running for office.

Broockman and Skovron use these results as a starting point for speculating on how constituents can control their representatives, given these systematic biases in the representatives understanding of opinions among their constituents. As I understand it, this approach fits into a large question within political science, as studied in the United States: How can democracy work even as good as it does in the United States, given the widespread ignorance of the most basic facts about the political system on the part of populace in the United States, including voters? Broockman and Skovron have added a new question: How can democracy work in the United States, given not only ignorance among the populace, but also systematic ignorance on the part of elected officials?

I would like to suggest two hypotheses for explaining these results. First, I suggest legislatures are accurately reflecting the views of their constituents, at least those constituents who matter. Martin Gilens finds that only the policy views of the rich influence what policy gets implemented, at least on the Federal level. Andrew Gelman has shown that the rich tend to be more reactionary in their views.

Second, I would like to suggest that norms of politeness in the United States interacts with conservative minds such that conservatives are systematically underexposed to liberal views among their constituents. I draw on Jonathan Haidt's work here. In some work, he defines five dimensions of moral intuitions:

1. Harm/care
2. Fairness/reciprocity
3. In-group/loyalty
4. Authority/respect
5. Purity/sanctity

(Quite a bit of literature exists on the different cognitive styles of conservatives and liberals. Liberals tend to have more activity in the anterior cingulate cortex, and conservatives tend to have a more active amygdala.) Liberals tend to worry more about harm and fairness, while conservatives equally emphasize all five dimensions.

My hypothesis is that conservatives tend to hear those articulating liberal, or even more left views, as being rude. If you are not comforting the comfortable, these days, you are branding yourself as not a member of an in-group that conservatives are loyal to, showing disrespect for our elites, and demonstrating personal impurity. So whether or not they understand liberal views, conservatives are unlikely to perceive such views as any more than eccentricities.

I suppose one could test my first hypothesis by comparing politicians' estimates of their constituents' views with the actual views of those constituents in the top 10% or 1%, by income or wealth. I'm not sure how one would empirically assess my second hypothesis, relating norms of politeness to political views. However one did this, I would think my second hypothesis would apply in a more extreme fashion to rural districts, as compared with urban districts. I do not know how this would apply in suburban districts.

I've probably made my usual share of spelling and grammar mistakes above. But I get to conclude this post, as if it were a journal publication, not a off-the-cuff blog post. More research is needed.

References

• David E. Broockman and Christopher Skovron. What Politicians Believe About Their Constituents: Asymmetric Misperceptions and Prospects for Constituency Control, Working Paper (3 March 2013).
• Andrew Gelman. Red State, Blue State, Rich State, Poor State: Why Americans Vote the Way They Do. [I HAVE ONLY READ PAPERS SUMMARIZING THIS]
• Martin Gilens. Affluence and Influence: Economic Inequality and Political Power in America, Princeton University Press (2012). [I HAVE ONLY READ A GILENS PAPER, NOT THIS BOOK]
• Jesse Graham, Jonathan Haidt, and Brian A. Nosek. Liberals and Conservatives Rely on Different Sets of Moral Foundations (2009).
• Jonathan Haidt. The Righteous Mind: Why Good People are Divided by Politics and Religion, Vintage (2013). [I HAVE ONLY READ PAPERS SUMMARIZING THIS]